I'm pretty sure that there's a lot more you can do with 限る than there is with 限定する, (When I looked up 限る, I saw a lot of different usages, such as it meaning "is the best," or "nothing is better than," in different contexts). But I'm trying to figure out which one is more commonly used for "to limit" or "to restrict." When should I use 限定する? When should I use 限る? Is one more formal than the other? What's the nuances here?
This may be incorrect but I have a hypothesis.
限る means “someone or something limits” 限定する means “someone limits”. 限定された時間 is only used like when a doctor allows a patient to go out in limited time. 限られた時間 is also used in that way and like when a person is too busy and has little time to listen to his favorite songs.
But 限定的 definitely has the meaning “someone or something limits” for example この方法は限定的な場面でしか使えない.
This is a bad answer that doesn’t have a definite conclusion. But I hope this helps you a bit.
限定する is usually used to describe the action to limit something. 限る (of this meaning) can also be used for the action but is more about describing the state of limitation. No one is more formal for this verb form.
申し込みを一人一枚に限定する means "To limit the application to one per person", and
申し込みは一人一枚に限る means "The application is limited to one per person". They are not identical in this case.
Also, 限定する only means to limit or to restrict a whole thing to its partitions.
限定された時間 is only used for cases like "this door is opened only from 7 am to 5 pm (partition of a day)", but
限られた時間 can also mean "the time left (for someone is going to die)".
But jumping out from the verb form it's another story. "limited to A" can be A限定 or Aに限る and are quite identical. You should use their correct form respectively in the sentence since they are grammatically different, though.
In fact, 限定 is widely used in the form of ～限定.
数量限定 (limited quantity),
本日限定 (only today),
限定販売 (limited distribution),
50部限定 (50 pieces only) etc. Do not be confused when you see this form.