Both are normal German words. They are not specific to one region. Zusammen is an adverb and gemeinsam is an adjective. There are some contexts where the two are interchangeable but there are others where only one of them is possible.
If your question were motivated by the belief that the two words were competing with each other and that only one of them were used by any one speaker, you would be mistaken.
Name the example where you cannot use gemeinsam and one where you cannot use zusammen. And if you can explain why is it so from synchronic point of view. I'm not interested in history of the use of the term, just why yes and why no.
If zusammen is part of a verb they are not interchangeable. compare:
"zusammenbauen - to assemble" vs "zusammen/gemeinsam bauen - to build together"
"Lass uns das gemeinsam/zusammen zusammenbauen - Let's assemble this together"
"zusammenlaufen - to converge (rivers aso)" vs "zusammen/gemeinsam laufen - to go/walk/run together"
"Lass uns zusammen an den Ort laufen, an dem Rhein und Main zusammenlaufen"
"Let's walk together to the place where Rhine and Main converge/merge"
As adjective you can only use "gemeinsam" in many cases:
Das gemeinsame Spiel der Kräfte. (*"Zusammene" does not exist.)
Wir kamen in Berlin zusammen. = Wir trafen uns. (Here "gemeinsam" does not work, because "zusammen" includes a direction and it is part of the verb.)
Wir kamen zusammen nach Berlin. = Wir kamen miteinander. (Here you can use "gemeinsam".)
Nothing to do with history. An adverb cannot take the position of an adjective. E.g., you cannot say "der zusammene Urlaub" instead of "der gemeinsame Urlaub".
There is some overlap because "gemeinsam" can also be an adverb. In German a derived adjective takes a form indistinguishable from the predicative form of the adjective from which it is derived (E.g. "Er ist langsam"="He is slow"; "Er geht langsam"="He walks slowly"). This might have confused you.
As you can see from Hutschi's and Küstenwache's examples, there are also semantic differences.