They were really alike...I was reading somewhere that originally Andy was going to be like Lucy, always getting into odd predicaments and needing to get out of them, but they decided that wouldn't work, so instead they made the other characters get into the problems, so that he could come and fix them...you have basically the same thing with Kermit.
Yeah, you can tell a difference in Andy's demeanor throughout the first season, and the following seasons of the show: throughout the first season, he's a lot more of a quirky character, with his country-fried retelling of stories (like telling Opie about Romeo and Juliet), or gently teasing Barney or Ellie about something, but afterwards, he's completely the straightman, or as they would say on TV Tropes, the Only Sane Man in Mayberry, while everyone else were the ones with the quirks, whether it was Barney's overzeal, or Otis's drunken stupor, or Gomer (and later Goober) being the simple-minded yet well-meaning bumpkin, or the nutty Ernest T. Bass, etc.
I understand that even though he was never credited as a writer, Andy pretty much mapped out every single episode of the show, or at least had a say-so in how the episode should play out and such... I remember one anecdote from Ron Howard about there was a line he supposed to say as Opie, but he felt it didn't seem like the right kind of line a seven-year-old would say, so Andy asked him what he thought the line should be; they used it, and he later told him how surprised he was that they actually used the line that he (a seven-year-old) suggested, to which Andy told him because, "It was the only one that was any **** good".