If you clear all the alt world puzzles, you can make all Mythical Pokémon appear permanently. Also you do not get to use the Primals. I have completed the Primal Groudon Mural in my game and all it does is say "yay you completed it in <time>", give you its height/weight/etc, and give you the option to do it again if you want (you can take it at any time) to see if you can get it done in a faster time.
I have finished all the main puzzles myself bar random mythicals (Need them to finish Primal Kyogre) so I am going through the Alt Worlds now (Only 300 Picrites to unlock all alt worlds for those standard worlds you have completed). For the money you pay out for it, there is a huge amount of puzzles you can do. I still have not decided to max out the EN gauge just yet (have it at 400) for the sole purpose of keeping myself from losing track of time for good though I should do it eventually.