The SID and Paula have NOTHING in common grrrr. I hate it when people compare the two, it's like comparing KFC to a Whopper meal. Yeah both are food/soundchips but the ingredients that make them up are nothing in common.
The SID is really an analogue synth on a chip, and the only spiritual successor I can think of to SID is the Ensoniq sound chip in the Apple 2 GS (same designer, Bob Yannes). It has many many features and effects for the time, and if you don't like its sound at all then you just don't like anything like that ie early 80s UK synth pop sound because it is the same technology really, a simple mono analogue synthesizer. Me I love it, not all tunes nope, maybe only 5% of the entire total of 100,000s of SID tunes out there, but 5% of that is more than all the commercial music tracks ever produced in the world that I like so. Anyway the point is the unique features borrowed from proper analogue synths not present in lesser chips like YM8912 or Pokey were key and well the diversity is clear, Hubbard tracks sound nothing like Galway tracks which sound nothing like Whittaker tracks....what I mean is you can't even tell those three composer's tunes are even being played on identical sound hardware. And listen to Cinemaware's game soundtracks too, very nicely interpreted from Amiga, far superior to the ST version aurally.
The Amiga had 2 stereo 8 bit DACs, that's it, some simple fixed freq muffler, sorry filter, with some volume controls. Amiga doesn't really have a unique sound, sure there were plenty of rubbish MODs in the 80s, but that's because everyone was using the same rubbish samples from crappy late 80s cheesy or rap music *cringe* but there are plenty of MODs that are very lovely and delicious indeed. Revelations slideshow is one that comes to mind from Crypto-burners, and as for games well Super Stardust is probably the best game soundtrack in the universe (and always will be!). But then these were done in the 90s as ambient music was getting a hold and rap crap wasn't shoved in your face at every corner TFFT. The Amiga's strength is the fact that basically you are given 4 DMA sound channels with AM/FM controls and off you go...do what you like, even mix them in software to make games with 6 channel sound total etc.
Both are great sound chips, both for VERY different reasons. Both have some funky tricks to punch above their weight like sample playback on original SID being quite good (listen to Arkanoid soundtrack or BMX Kidz) and Paula has a 14bit sample playback routine
You don't see FM based Soundblaster cards with patches to play PCM type samples on PC do you?
Don't think the original Mac had sound, and ST had the same inferior Yamaha chip as 8 bit Sinclair/Amstrad units. NES had rubbish sound, SNES sounded a bit General MIDI extra cheese on the side style too. So is there any wonder compared to the competition that of all the mass market machines people generally only remember the two from Commodore!
(I guess what I am trying to say is even though both are excellent if you haven't got a clue where to start then it is quite possible to listen to hours of crap SID or Paula tunes sure, but then that's your fault for not asking people in the know what composers to start with)