There's some difference between 수영을 and 헤엄?

Question

Answer

수영(水泳) is sino-korean word. but 헤엄 is native word. Both of all means swim.

헤엄치다. 헤엄을 치다.
우리 여기서 저기까지 헤엄쳐서 가보자.

수영하다, 수영을 하다
우리 여기서 저기까지 수영해서 가보자.

Answer

When you wanna use the word "수영" the verb that goes with is "하다"  
"헤엄" is used with a verb "치다"
헤엄 can be any form of movement in the water, whereas 수영 can be in accord with established forms.

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