In grammatical terms, 楽しい is an adjective whereas 楽しみ is a noun. You can think of 楽しみ in the sense of "enjoyment," though the Japanese often say it to mean "looking forward to." Off the top of my head:
「楽しみにしています」
"I'm looking forward to it"
「明日パーティーがあるの」
"There's a party tomorrow"
「うん、楽しみだ」
"Yeah, I'm looking forward to it."
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