My limited Google-fu shows both appear to be in use. Google Ngram is very puzzling, as it shows the two terms switched popularity positions. What are the differences between these two terms?
If you compare British and American corpora, the split is more distinct. While swimming team was the more common term in both a century ago, swim team has soared in popularity since the latter half of the 20th century, and is the far more prevalent form in the U.S.:
Still, it's not difficult to find swimming team in respectable American sources like Swimming World magazine, and few would object to swimming and diving team over swim and dive team (many high school and university athletic departments combine the two sports into one program). In other words, there is no difference in meaning between a swim team and a swimming team, only in preference or familiarity.
I don't think the swing can be attributed to any one particular cause, and it isn't applicable to other sports or word pairs. Swimming coach, for example, has been more resilient in AmE vs. swim coach:
The -ing remains safely ensconced in many sports: there are far more cycling teams than cycle teams, and I do not think boxing, fishing, or cheerleading are in any danger of changing, for example. On the other hand, I'm perfectly fine with ski instructor, hunt club, skate coach, or snowboard helmet where some might insist upon the progressive forms.