You will not be able to claim the other park areas as you did with the raiders.
You will still be able to explore and do all area-specific quests - as soon as you go near one of the areas, the quests will pop up. The traders will however not become a faction, so siding with them is quite a bit of an anti-climax.
Siding with the traders, you will not be able to claim the areas with flag poles as you do when siding with the raiders. Neither will you get the loot chest that the raider gang would offer you when you hand them the area.
Various minor cosmetic details will be affected depending on which side you pick: you will get different trader dialogues if you side with them, and the Raider radio channel will go offline.
I would strongly recommend siding with the raiders for the first play-through, as you'll get a whole lot more content that way. It is clearly how the DLC is meant to be played - siding with the traders isn't very interesting. It will give you one big, difficult shoot-out killing every raider in Nuka World - probably the hardest combat challenge the game offers. But when that's over, that's that.
Other Nuka World quest lines like the hubologists, the hidden Cappy, the park medals etc are not affected.
Notably however, The Minutemen and raiders are nemesis factions. So your relation with Preston will get screwed up unless you side with the traders - he will refuse to join you as companion until you do. Similarly, you will become an enemy of Porter Gage if you side with the traders, so you will have to chose between Preston and Gage.