I'd say the main difference would be that; "Felicidades" is used to congratulate someone for a particular occasion (birthday, wedding, etc). "Enhorabuena" is used to congratulate someone for an accomplishment (promotion, award, etc)

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Felicidades is used mainly to wish a happy birthday and similar. Enhorabuena is as if you say "about time..." with a positive connotation - you can even include "felicidades" For example, Jessica got promoted after 4 years in her position, I'd say to her "enhorabuena, te lo mereces mucho. Felicidades!"