1. 한국에 가 봤어요.
2. 한국에 가 본 적이 있어요.
3. 한국에 간 적이 있어요. (가 적이 -> 간 적이)
(After the correction on #3) They all state a past experience as your translation says.
Korean past tense by itself (like 갔다 in predicative, and 간 in attributive adjective form) represents either a simple past action or present perfect in the sense of completion. So 그 사람은 갔어요 would mean "He went" or "He has gone", and never anything about experience.
-아/어/여 보다 is a helper verb that adds the meaning of doing something for trial or test. Its past past tense (가 봤다) denotes that the speaker has had certain experience, as in your #1 example.
-(으)ㄴ 적이 있다 ("There was a time I ...") literally states you had certain experience in the past. So it can be used like the English present perfect about past experience. Your #3 sentence is a case of this.
#2 is combining both phrases in one sentence (which is often done in real usage), so it has the clearest sense of the three.
Generally, -아/어/여 보다 has a lighter feel and is often used to ask about interesting experience, while -(으)ㄴ 적이 있다 is more often about straight facts.
For example,
- 한국 음식 먹어 봤어요? / 먹어 본 적(이) 있어요? Have you ever tried Korean food?
- 5월 25일 이후에 피해자을 만난 적이 있습니까? Have you ever met the victim since May 25?