I believe they both indicate the same thing,  but "on the flip side" is a more casual or informal way. - I hope this has provided you with an answer to your question. I'd be happy to assist you in developing a study plan if you'd like to practice speaking. Please have a look at my teaching profile and class offers and contact me if you feel I can help you. 😄

"Catch you on the flip side" is also slang for "see you later" :)

There's no difference in meaning. There's a difference in tone. "On the other hand" is good, normal, straightforward, standard English. "The flip side" is informal. Literally, it refers to the days of vinyl records, particularly the little 45 rpm disks that had one song on each side. You would play the first side, then you would flip the record over and play the second side. So the second side was "the flip side." Usually the song on the flip side was less popular. "'We Will Rock You' was the flip side of Queen's hit song, 'We are the Champions.' To peoples' surprise, the flip side became an even bigger hit." Figuratively, it can mean "the other side in a debate or argument."

"On the other hand" is more formal, and "on the flip side" is more informal. Hope this helps!