On the most simple level, it's probably because he's all old and stuff. Slapping around Sentinels is really a game for the young. Or Wolverine, who manages to get by on pure attitude, and also by being able to kill everyone in the world if they point out he's getting on a bit. Plus, Logan deliberately cultivates his status as an outsider. Banshee got stuck in his, and feels the need to prove to everyone (not least himself) that he was worthy of his status as an X-Man: that he is no less a hero because he'd rather smoke a pipe than do... Christ, whatever it is that young people do now. Play Pokemon? Yu-Gi-Oh? Are they still roller-blading, or what?
I'm not sure anyone listens, though, which is a Hell of a thing to take when your mutant power is a sonic fucking scream. Banshee just keeps being ignored, not through malice, or even disrespect, but just because the young tend only to really see the young.
Not all that long after he joins the team (another recruit hired in the hopes he can feed Krakoa its own living-island arse) he finally gets his chance to justify his place on the team by saving the entire Japanese archipelago from the deadly powers of an Earthquake Gun (it probably had a better name in the original, but maybe not by much). He pulls it off, too, but in the process he loses his sonic scream. It's a pretty bitter pill to swallow when you finally get to prove your skills pay the bills only to have them immediately snatched away.
This, though, is just the most literal example of the problem that has seemingly followed Cassidy for years; that he is desperate to prove himself, but he has no idea how to do it, or even what he wants to prove himself as.
Actually, that maybe isn't it, exactly. It's more that Banshee wants to prove himself in seven different directions at once, all the time, and there's no reason to believe those directions are complimentary, or even not perpendicular. Xavier spends so much time thinking about the big picture he can be a bastard to deal with on an individual level. Wolverine is so completely fixated in the now that he's afraid to think far enough ahead to decide what he'll have for dinner tomorrow. Neither of these states of affairs are in any way balanced, obviously, but at least they exist on one end of a scale. Banshee seems completely determined to hit both extremes simultaneously.
Consider the path his life took. Rather than spend his time bumming around his family's County Mayo castle with his cousin Tom, and playing hide and seek with the local leprechauns (I am not even faintly kidding), he decides to join Interpol  in order to do some good (where "some good" proves ultimately to involve letting first Magneto and then Omega Red manipulate him like a sock puppet, but that's not really the point). He spends some time on secondment to the NYPD for a while, as well, then ultimately joins the X-Men. Following on from that, he becomes headmaster at the Massachusetts Academy (at that point essentially a separate campus of Xavier's School For Gifted Youngsters), and finally forms X-Corps with the goal of policing the mutant population across the globe.
All of these activities share a common theme: the desire to improve the world. Maybe not by much, but the big picture is key. Or at least, it should be. Banshee can't keep his eye on the prize any more than Wolverine can. On multiple occasions he defies orders in order to do what he thinks is best, chasing people all over the world (yes, they're serial killers, but what about the paperwork, Sean; WHAT ABOUT IT?) or breaking cover to help out a harassed woman.
Now, I'm not saying that we should let serial killers run wild as nature intended, or that saving women from street thugs is beyond a cop's remit, but it is indicative of Banshee's central problem, namely that he wants to be both sides of the coin at once. Given all this, putting him in charge of young teenagers and telling him "Look after this kids but also train them to save the world" is pretty much the dumbest idea Xavier has ever had, and he was all "Give me some of that alien bird poontang, willya?"
It's this inability to separate the micro from the macro, and his unwillingness to understand the difference between what he can and cannot solve, that leads to so many of the disasters in his life. The first of these occur while he's away on an undercover mission for Interpol, during which time his wife Maeve is killed in a bomb attack. He blames himself for her death, since he wasn't there (dumb), but then tries to transmute that guilt into anger by attacking his cousin, who was there, for not helping either (dumb and cruel since Tom loved her too, and had stepped aside in order for Sean and Maeve to be happy ). Given this extreme bout of dickishness, Tom responds by not mentioning that Maeve had had a daughter whilst Sean was gone, instead he raises her as his own.
When he isn't losing family members in droves, Sean's refusal to toe the line is causing problems elsewhere. Targets are lost and fellow law-enforcers are killed as Sean tries to prove that he can save everyone all the time, no matter how idiotic that idea clearly is.
It never works, of course. People still fall through the cracks. At one point Sean is captured and mind-controlled into helping an alien species from Sirius try to destroy all of humanity. Each set-back just gives him one more thing to prove himself against, and yet the more mistakes he racks up, the more desperate and thoughtless his attempts to combat them become. He's like a gambling addict on a losing streak, constantly increasing his stake in the hope he can win it all back. Or maybe it's closer to being in a relationship, where you're so desperate to make sure you don't make any of the same mistakes that you don't notice all the exciting new ones you're making instead. Banshee gets so caught up in being a superhero that even when he discovers he has a daughter he isn't sure he has the time to handle her, to the point where when she loses her similar powers to his in a similar way to him, it still doesn't make her interested in him as a father figure. He desperately loves Moira MacTaggert but is frequently too busy to be with her, to the point where he is absent so long that when she dies nobody bothers to tell him. 
This is the point at which the wheels really come off the wagon. Well, something comes off the wagon, at any rate; since Sean chooses to embrace centuries of ingrained stereotypes and booze his way to happiness. This, of course, comes only weeks after one of his charges was killed in a bomb attack (dude, this guy is so all about those he cares for exploding) and his decision that he wasn't teaching them well enough. And who would have thought; turns out unspeakable grief and enough booze to make Oliver Reed queasy don't really do much for your powers of imparting knowledge . Chamber graduates to the X-Men (and immediately becomes shit, but we'll get to that in SS v X #45 ), and the others go their separate ways (some return to the fold later, others do not, and poor old Skyn gets quite literally crucified). Stung (though I doubt he would admit it) by being abandoned, and horrified that Moira not only gave her life to save mutant-kind but that apparently he wasn't on anyone's phone-tree during the crisis, Banshee decides it's time to Rambo up, and he forms X-Corps: a massive, international organisation with the self-appointed remit of policing mutants.
Naturally, Sean approaches this new task, the glorious work that will undo all his past mistakes, by simultaneously focusing on the bigger picture (to the point where he has no problem hiring villains to help out, since he has them "pacified") and the personal (persuading three former members of Generation X to join in). It's that last part, understandable as it is, that puts the final nail in the coffin. Cassidy is a hero, there is no doubting that; he has saved the lives of hundreds of thousands, but his insistence on throwing his former charges (who he once described as "like his own children") into battle alongside former murderers seems to finally prove that somewhere inside his head Sean is fighting simply to earn respect. Is X-Corps a name meant to honour his comrades, or to impress them? Is any of this anything more than proving he could have saved Moira if anyone had let him know she was in danger? Not to overstretch the metaphor, or anything, but does it really all come down to the fact that Banshee just can't seem to make himself heard? 
In the end, once again, he is so busy fixing the past he ignores the present. Mystique infiltrates X-Corps, and cuts his throat, severing his vocal chords and almost killing him. The organisation collapses, several of the "reformed" villains trash Paris (yeah, I know, but let's pretend to care), and suddenly Sean has one more mistake to atone for.
He never gets the chance. His voice has still barely returned when he discovers the truth about Vulcan and the other forgotten X-Men, and he dies trying to save the passengers on the plane that Gabriel Summers targets for no better reason than Sean is on-board, and he wants to delay his big reveal for a few hours more.
I get that not everyone gets a dramatically satisfying conclusion, but I wish Banshee had managed to save that plane. It would have been nice for his last act on Earth to not have been a failure. Of course, the flip-side to that argument is that his heroism lay in the fact that he tried, despite his injuries, and against the odds. Perhaps that's the lesson to take from his death, and perhaps if Banshee himself had learned it, his life might have been the better for it.
Next time: we explore whether or not I can entire article on Storm without lapsing into a coma, and consider whether that might not be the better option anyway.
 I don't know whether this has even been explicitly addressed, but I always assumed Interpol was used by the writers to avoid any possible suggestion that Banshee had taken sides in "the Troubles".
 Sure, the man who became Black Tom is responsible for his own actions, the crime-sprees and the killings, but it's worth noting that some or even all the good Banshee does with Interpol is erased by treating his cousin so badly (as well as beating him up) that he ends up swearing revenge by any means necessary.
 Actually, this is only partially because he wasn't sufficiently attentive, it's also about how no-one in the X-Men ever seem to notice it when he's gone. No-one hears the screamer, y'know?
 Remember that old saying about "Those who can't, teach."? I'm wondering how fair it is to say that Sean is a pretty good case in point.
 If anyone cares, we're currently about a quarter of the way through this ludicrous exercise. I shall probably move onto the New Mutants afterwards, or possibly just collapse and die.
 Once again, it's interesting to compare "our" Banshee with his Age of Apocalypse counterpart, who spends his time trying to hide his cowardice rather than prove his bravery. Dragged out of retirement by Magneto to fight in the final days of the war, Banshee gives his life to kill Abyss, pointing out that the world needs one less of Apocalypse's Horsemen far more than it needs one more of him.