I think they mean exactly the same thing. English words seem to follow a regular pattern of evolution, in which compound words tend to lose their hyphens over time. To me (as a US speaker) "heart-warming" looks slightly old-fashioned. Where did you see it? I have a little bet with myself that it was published before 1920. Added: Having checked the American Heritage dictionary and OxfordDictionaries online, I lose my bet with myself. Rene is right. "Heart-warming" is British, "Heartwarming" is US. Make your choice accordingly.

while different dictionaries seem to give slightly different definitions for the two, I think they are the same and that the difference is just "heartwarming" being more US and "heart-warming" British English.

I've only used and seen heartwarming. Heart-warming looks like something from the American Victorian era.

They mean the same thing, in my opinion it looks better to write it "heart-warming” than “heartwarming” but that's just an opinion.