There is no true difference, but you could say that 헤엠하다 is more like the actual serious practice of swimming. 헤엠 on its own means "(a) swim stroke" and it's used like "헤엠 잘치는 사람" (a good swimmer), "헤엠치지 마시오" (No Swimming, like on signs). 수영하다 also means to swim, but in 수영하다 the concept of swimming is a bit loose, and you can use it as "bathing" as well, like "풀장에 수영하러 가자!" (let's go for a swim/bathe in the pool!). I'd say su is more commonly used, unless you're a professional swimmer where you'd maybe hear he often as well. I wouldn't say to my friends "헤엄 치러 갈께" (i'm going to go for a swim/take a swim) because that'd make me sound like I am going to go practice professional swimming, when I'm actually just going to loosely excercise swim or bathe. I hope this cleared it up a bit :-)

"헤엄" is a native word, meant "(of a person or animal) a movement or motion of swimming in the water (with arms and legs or its fins)", so "헤엄치다" and "헤엄하다" mean "to swim". The noun, 수영[물 수(水), 헤엄칠 영(泳)] is 한자어(漢字語), its meaning is "물에서 헤엄치기(to swim in the water)", so "수영하다" means "to swim". However, 수영하다 is usually used only with person, not with animal. for example, 물고기가 헤엄치다. O. 사람이 헤엄치다 O. 물고기가 수영하다 X, 사람이 수영하다 O.