"I want to swim" vs. "I want to go swimming" -- same thing. The difference is that strictly spoken in the first, you desire the action itself (as in, "I want to swim, not just only lie in the sun", whereas in the second, you want go somewhere and do it ("lets go to the Schwimmbad").
But there's more to it, really. The
INF + gehen construction is used in all kinds of situations to express that the action in
INF is executed after moving somewhere else. This is strictly a construction, since you can't just use any verb of motion, like *ich laufe schwimmen. It also has a certain future aspect, as you can see in English, where the same construction "going to X" has evolved into a proper future tense.
You can pack all kinds of phrases into it, actually, but note that some of them have quite an informal sound (you wouldn't use them in writing):
- Ich gehe laufen: "I go jogging"
- Ich gehe sie holen: "I go and fetch her"
- Ich gehe ihm helfen: "I go and help him", quite informal
- Ich gehe etwas kochen: "I go and cook something", somewhat (?) informal
I find it difficult to find a general rule about the formality other than a tendency of complex phrases being more informal. Only infiniteve + gehen is always OK, though.