Yes, there certainly are tradeoffs. It could be a long response, but a couple of points, for a start: A softer leather glove, like the various deerskin or kangaroo skin, will generally be softer, giving you better "feel" of the string, and what I think of as string/nock control. Better sensitivity and dexterity for other associated tasks as well. That feel also means less "protection" from the string. Also may tend to have a little less crisp release, and may be more prone to grooving. (All relative to the A.L> nylon tipped glove you shoot, which likely has plastic inserts also). The thinner, softer leathers will also give you a better feel for your anchor point.
A heavier/stiffer, all leather glove - and they vary within this category too - is sort of in between. More protection, may have a little cleaner release, more protection. A little less feedback on your anchor. Slower to groove than the softer leathers.
Just to get things rolling; I'm sure more responses will start showing up soon.