Begrudge: resent the pleasure or good-will of someone. Grudge: to feel unease or resentment.

Begrudge is a verb, e.g. don't begrudge me my good luck/he begrudged the fact that she was with someone else. Grudge can be both a noun (e.g. he held a grudge against her for years) or a verb (e.g. he grudged paying so much).