According to my Longman dictionary, gain experience and get experience seem to mean the same:
gain/get experience: The programme enables pupils to gain some experience of the world of work.
But I have also found gather experience and acquire experience (source):
We now have to gather experience.
These projects will set their imagination in motion, and will help them acquire useful experience.
How do these versions with gain, get, acquire and gather differ? I am also interested if they mean the same but sound slightly different to native speakers.
Gain adds to get the notion of something profitable or desirable (which is why we can get sick, but we don’t gain sickness.) So, saying that an internship will help you gain experience implies that the experience you get might be useful down the road, such as on a resumé for a future job.
Using gather or acquire in conjunction with experience seems more suitable for corporate applications than personal endeavors. In other words, I can imagine a manager saying something like, “We need to acquire more experience in database applications before we put in a bid for projects like that.” However, if someone is talking about personal experience, (as in “I want to go to a third world country before starting college; I think I would gain some valuable experience”), then I wouldn’t recommend substituting gather or acquire in that context.