Are they the same, meaning to make people feel emotional?
The standard phrase or idiom is tug at one's heartstrings:
Also, NOAD defines tug thus:
Thus, when something tugs at (pulls at) your heartstrings, it deeply affects your feelings in some way (a prick, a pang, etc).
"Tug (or 'pull') at one's heartstrings" is an established idiom. "Tug at one's heart", is not, though it is a perfectly valid expression, with the right meaning.
Your question is slightly ambiguous, because of the word "make": to be clear, it is the thing that elicits compassion (which might be a person, an action, or an event) which "tugs at your heartstrings". You wouldn't use it of the person who was doing an action, unless they elicited compassion in themselves.
"Heartstrings" is pretty well obsolete, apart from this expression (the OED has no examples since 1896, though the entry was updated in 1989).