Why do I collect rings in this game? No matter how many I manage to rack up, I lose almost all of them if I get hit once and its almost impossible to me at least to get through a level without getting hit. The only reason I found is that I can survive a hit, but thats the same wether I have 10 or 200.
100 rings = extra life, 25 rings at a checkpoint lets you subject yourself to Blue Sphere stages, and really, the more the merrier. You get more bonus points at the end of a level the more you have (and every 50,000 points also gets you an extra life). They're easier to pick up if you get hit while carrying a lot of em.
Classic Sonic is pretty arcade-y, and the mindset behind them is usually that players will replay levels, learn layouts, and get a little better each time. Mania even has a Cool Bonus now that starts at 10,000 points at the beginning of every level. Getting hit takes off 1,000 points and dying zeroes it out.
So just think of it as an arcade game, bigger numbers are nicer just 'cause. If you're just trying to get to the end and be done, they don't matter too much, but they're nicer to have than not. You don't have to go out of your way to grab every ring you see, but they can also clue you in on alternate routes. If you're worried about losing them all, keep an eye out for the Blue Ring item box. They make it so the rings you lose on hit are huge and worth 20 apiece.