Many cars and trucks can be bought with a "towing package", which usually includes both increased engine and transmission cooling capacity - there's extra strain and wear throughout the drive train, but the biggest problem is when the engine or trans overheats, which can cause major damage right then.

both

I don’t think it’s an either-or question.  You can get a different answer depending on the rig you’re talking about.

For example, let’s say you’re talking about a full size truck towing a 5000 pound trailer.  If the truck has a turbo 4, towing that much weight on a regular basis could cause a great deal of wear, whereas the same truck with a V8 might hardly work up a sweat. 

Similarly for transmissions, it also depends.  For example, GM’s 6 and 10 speed transmissions are pretty solid, but their 8 speed reportedly does not stand up well to towing.

With all that out of the way, I would say, across all makes and models, the greatest relative strain from towing seems to be on the transmission. I’ve seen a number of people who tow regularly need the transmission rebuilt/replaced prematurely with the engine doing fine.

I would say more the trani if an automatic. Increased load and slippage load would heat it up. Long up grades the worst for both trani and engine.