Hmmm... that "rule" works for some emotions, but not all. In this case, both expressions are the same. It depends, as Poli says on the speaker's preferences.
Personally, I think that we tend to use more often the specific verbs when available than the equivalent expressions, because the other constructions sound a little more colloquial:
Me entristece = me da tristeza = me pone triste
Me alegra = me da alegría = me pone alegre
Me enoja = me pone enojado (we don't say "me da enojo", but to express the same emotion we say "me da rabia")
Me avergüenza = me da vergüenza (but not "me pone avergonzado")
There are other emotions that won't accept another construction, but their dedicated verb:
Me sorprende (we don't say "me da sorpresa" or "me pone sorprendido")
Me complace
Me entusiasma
Me relaja
Me preocupa
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