I think the answer by Super Cool Handsome Gel Boy makes sense, so let me supplement it. In my opinion, 아/어 보다 could be translated as 'try ~ing' rather than 'try to'. But I am not so confident in my English, so I need English speakers' help.
According to my understanding, 'try ~ing' is used when you are doing something already or try to do something expecting some future influence. For instance,
I am trying going vegetarian. (I have already stopped eating meat for a while.)
Try being quiet. (Be quiet and keep quiet.)
If this understanding is correct, I can give an explanation. Two expressions would be translated as
ㄴ적이 있다: there was an event that ~ (better translated as 'have ever ~')
아/어 보다: try ~ing
If you are using the second expression in the past tense, you are saying that the action was already taken. So two sentences are basically the same in this case and can be used interchangeably. Of course you can combine two expressions.
- 아/어 본 적이 있다 (have ever tried ~ing).
I think the third expression is most common in Korea because it sounds somewhat polite. The following might be an answer to Rathony.
- 서울에 가려고 했는데 못 갔다. (correct)
I tried to go to Seoul but couldn't.
- 서울에 가 봤는데 못 갔다. (very weird and not recommended, could possibly be used by some comedians because 가다 could be interpreted as either 'go' or 'reach')
I tried going to Seoul but couldn't.
- 서울 방향으로 가 봤는데, 다리가 끊어져서 도착하지 못했다. (fixed)
I tried going toward Seoul, but couldn't reach it because the bridge was broken.