The votes still count against badge mechanics like Nice Answer or Good Question. You just won't be able to get anymore reputation.
There are some mechanisms that circumvent this rule
- You mentioned the "Accepted Answer", which is probably the most important one. You always get 15 reputation if an answer of yours is accepted.
- The second one is accepting an answer yourself. This yields 2 reputation.
- The last option is getting a bounty. If you answered a question that has an open bounty and you get the bounty either fully because the person that placed the bounty manually awarded it to you or half because the person forgot and the system automatically distributed only half of the original bounty you can go beyond the 200 reputation.
One last thing to consider: when you reached the 200 rep for a day and were downvoted you can still get the 2 rep back by another downvote. So it's not like you are "cut off" for the day. The system checks with every rep income whether you are at the limit for "normal" rep gains up to 200 per day.
Think about it like two different buckets:
One is the normal bucket. This is the "200 rep" limit per day. Normal upvotes on your questions and answer go here. Rep gains through approved edits from posts, if you are under 2k rep, and approved edits to wikis go here, too. Downvotes on your posts and downvotes you cast on other peoples posts go here. You can fill the bucket in whichever way you want. But you can't go above the 200 rep. And once the bucket is full any excessive rep is lost. It is not kept for later use in case your bucket empties a bit throughout the day.
The second one is the special bucket. It is so big that you don't need to worry about filling it. SE makes sure this bucket is as big as you need it to be. Here you get the rep from accepting answers, from your answers being accepted and from getting a bounty through manual and automatical distribution.
The badges regarding the maximum amount like Mortarboard count those two buckets together.
See here for information on Meta StackExchange: What is the daily reputation cap and how can I hit it?
There are a few strange special cases that mainly involve un-downvoting. See here for example: Strange rep cap behavior: how did I get 201 for the day?. Basically if you downvote, reach a net 200 rep for the day and then un-downvote or the downvoted answer gets deleted you regain your spent reputation. Think about it like the rep comes from any of the buckets, preferrably the normal one, when you downvote and the regaining is put into your special second bucket. But those cases tend to be rare, so don't sweat it if you get weird numbers like 201 or 216 reputation on some rare occasions. This can even lead to something like 217 if you downvote multiple things and undownvote them, as the one undownvote rep is always put into your special bucket. Theoretically you should be able to raise the max score to 230 (+ normal special bucket rep) with this by using up your 30 answer-votes on downvoting answers, then getting some rep, editing all the answers and undownvoting them. Maybe the mod tools would show this at some point. I haven't tried it, but it should be possible in theory. Quite a lot of effort for a meager amount of imaginary internet points though... Writing more answers and hoping one gets accepted sounds like a better plan.