If you put aside the "influence" factor (which I think is debatable in either direction), Batman Begins is just pound for pound a better movie in my book. Better story, better script, better cinematography, better score, more of an emotional pull, a strong protagonist with a clearly defined goal and a fantastic journey throughout the film.
To me about the only thing the films have in common besides being superhero films is they both employ great ensemble casts. It's almost like comparing a drama to a comedy to me, because The Avengers' biggest victory was striking a fun/comedic tone that made the idea of all these worlds colliding in one film more palatable. You take the comedy away from that movie and it becomes something completely different and probably doesn't work nearly as well. Now, I give major props to Whedon for 'cracking the code' on something that most thought would flop, but at the same time this issue just leaves me feeling like there isn't a whole lot to the movie beneath the surface. I think even Joss has said that he doesn't think it's a great movie, just a really fun ride.
Begins, while perhaps not an all-time classic (and IMO the "least" impressive of the trilogy), is pretty solid through to the core, even with a few notable flaws, and has held up over time as a good piece of filmmaking and arguably the postmodern template for the idea of the "reboot" which didn't even exist in our cinematic lexicon prior. For me though, what I love most about the film is how tactile it feels...incredible location shooting, massive soundstage sets, miniature work, and CG in the most unobtrusive way possible. I think both films harken back to the 70s/80s golden age of the blockbuster, although with Avengers it's more in the spirit and whimsey of it and with Begins I feel it's an attempt to do a modern version of that, in the sense that it utilizes more of the filmmaking methodologies from that era, but also in that it has a brave-faced earnestness about presenting a larger than life figure within a believable, albeit heightened reality ala Donner's Superman.
I don't think that was any attempt to emulate that Spider-Man line. I doubt Nolan's even seen Spider-Man. It was simply a line that sums up an inner strength in Bruce that when he fails at something he learns from it and recovers from his failure.
I agree that the line likely has absolutely nada to do with ripping off Uncle Ben, but I'd wager that Nolan's seen Spider-Man. He's a fan of big blockbuster films, and we know he saw X-Men due to Jackman's comments on how Nolan worried that Singer had beat him to the punch when he first saw it. Besides who
didn't see it back in '02? The film was massive! Also, Jonah references Spider-Man in the Trilogy Boxset documentary.