Not much. "Paljon" means lots or much, so "paljon onnea" could be translated as "lots of luck", while "onnea" is the partitive form of luck or happiness (luck/happiness = "onni"). Both expressions can be used when congratulating someone (for their birthday or some other joyous occasion), or when wishing someone for luck or success in some endeavour, like an exam. The shorter version might be a tiny bit more familiar, but I don't feel there's much difference there.
I think "best wishes", while not wrong, is in most cases inaccurate as a translation for "Paljon onnea". Best wishes would most often be perceived as closer to "Kaikkea hyvää" or "Hyvää jatkoa" whereas "Onnea" almost invariably is a congratulation for something unless the purpose of the congratulations are specifically mentioned to be something in the future. "Best wishes" would be applicable in eg. "Paljon onnea [uudessa työssä/uuteen työhön]".
But as I said, it's not wrong, it's just not how most Finns perceive it.
In English, the time frame should be considered. "Best wishes" generally implies an occurrence in the future, i.e. you are about to start a new job, embark on an adventure, take your driving test, etc.. Congratulations are given when an event is now taking place or has recently done so, (e.g. I have not seen you since you became engaged to be married). Cheer! is normally in the present - we are having a party now.
But "congratulations"? Doesn't that sound a bit... strange? I don't think I'd ever congratulate someone for being beautiful, but of course I could say e.g. "You look beautiful today" or something. :) That's "näytät kauniilta tänään", by the way.
But, in case you DO want to say "you are beautiful, congratulations" then you'd indeed say: "olet kaunis, (paljon) onnea".