I read this term shave cream in NY Times today. According to the results in COCA (Corpus of Contemporary American English), shaving cream (I also found this word in the dictionary) is far more common than shave cream in American English. I would like to know if there's any difference between these two terms.
As others have pointed out, both terms are correct and both refer to the same product.
However, don't use them both unless your goal is to specifically confuse your audience. When you are writing, use the term "shaving cream" as it is by far the more common and understood term. Public writing is done for the reader, not the writer.