Before I leave 1964 and address the musical changes both bands underwent in 1965, I'm going to show concert scenes for each of my previous two songs, with some commentary.
"Fun, Fun, Fun" in concert - Concert in March 1964.
"I Want to Hold Your Hand" - first US (non-Ed Sullivan) concert in 1964.
Lots of screaming kids in both.
Though, other Beatles concert footage I've seen shows the screaming at Beatles concerts was at a whole other level than just about anyone else.
Anyway, some commentary on both bands' early eras (pre-1965) ...
1) Both bands had their own "uniforms" for their concerts - the Beatles dressed in suits and ties, the Beach Boys in their famous Pendelton striped shirts. Not sure what, if anything, that could signify. It's tempting to say the BB uniforms reflected a casual California style with a dash of preppiness, while the buttoned-down Beatles represented a more staid and formal European sensibility. But we all know the Beatles' use of suits and ties was really at Brian Epstein's insistence and they probably wouldn't have done that if not for him, so as tempting as it might be to try to read something into their concert wardrobes, it probably doesn't really mean anything.
2) The Beach Boys came from an established American tradition (not very old at this point in time, but it was there) of rock-n-roll being Chuck Berrry R&B, Elvis Rockabilly, etc. Britain didn't have a strong rock/pop musical tradition at this time, so this gave the Beatles an opportunity to invent one ...
... As a result, with the Beach Boys we got an easily identifiable style of their early songs either being, A) an R&B, pure rock-n-roll style song, or, B) a ballad. There were some gray areas (there are always gray areas), but that was the general pattern ...
... The Beatles, coming from somewhere with little rock tradition, got to invent one. As we all know, their stuff emerged from Skiffle, which in the UK was a folkish-sounding, fast-beat style which might sound like something coming out of the Appalachian foothills.
It's also interesting to note that the Beatles used acoustic guitars extensively even in their early years, whereas the Beach Boys used them only rarely. Not sure how much, if at all, the Skiffle origins had any effect on the Beatles use of instruments, but being familiar with both groups' catalog, I think it's fair to say the Beatles frequent use of acoustic guitar had an effect on their sound and musical style. Acoustic guitars don't really lend themselves well to Chuck Berry-style hard rock tunes. On the other hand, if your musical inspiration/roots was a folk-style kind of music, acoustic guitars would be perfectly acceptable.
So the frequent accusation that the Beatles weren't really a "rock" band but were more of a "pop" band is somewhat correct, but that's in large part because they came from somewhere that didn't have much of a rock tradition or style yet. That said, I would argue that that, in itself, was a type of innovation; probably more than any other band, the Beatles blurred the distinction between pop and rock, while professing to be a rock band (though I suppose you could even say that of Elvis, so I admit my argument isn't so clear-cut). A lack of tradition can be liberating.
Anyway, I'll start on some 1965 comparisons later. This is when both bands really started to blur musical distinctions.