"The movie is enjoyable."
Both terms are fitting, they simply mean slightly different things.
재미있다/재밌다 - interesting AND/OR fun to do/ fun to watch/ fun to play/ fun to whatever
즐겁다 - happy, funny, positive, upbeat
A horror movie might be 재밌다 but certainly not 즐겁다 (might be nice to watch, but is rather not too funny and colorful).
I would argue otherwise. I feel you can have something that is enjoyable but not necessarily "fun".
Both invoke positive emotions, sure, but fun may depict physical engagement - e.g. playing sport, snowboarding, playing video games etc can be fun, however passive activities arent "fun". The movie was enjoyable, but the act of watching it wasn't fun (no physical stimulation), same could be said for a meal and eating it - the meal was enjoyable but the act of eating isnt fun.