Think of a press as a big slow hammer. The main difference, depending on die shape and available pressure, is that a hammer will tend to work the billet from the outside toward the center and the press will tend to work the billet from the center outward. This is the only advantage I can see if you're making patterned steel as it tends to close up any inclusions in the middle of the billet and squeeze the crap to the outside. This reduces (theoretically) the scrap rate in billet production.
I know smiths who use one or the other with good effect and swear by their tooling. I know smiths who have both. It's what tool you have and how you get to know how to operate it. I use hydraulics for welding, but I can weld in a power hammer too.
Forget anything hand operated, or with any lag in getting the top die into the steel in a quick hurry. Welding won't wait more than seconds. That being said, as good practice for the newbies, enough grunt in the press will make up for some temperature loss and you could weld at temperatures that critics would say were impossible. Give yourself a few years and you'll probably figure out what I mean.
Fast rpm electric motor turning a 16-22gpm hydraulic pump into a six inch cylinder on an H-style frame. These have been made in a C-style too (historically the first ones were former log splitters), but the H-style is more stable and less likely to bend over time. OTOH log splitter frames are cheap.
The idea that you don't need much force to weld a billet is very true. When I first "set" the weld in my press I barely squeeze at all. Then I rock and roll to forge out the billet and create as much movement in the random patterns as I need. The hydraulic press can be very controllable, but that's true of any forging tool. Dies make the work more efficient.
If I could have only one tool, it would be a power hammer. There's a lot more that could be said but this is the gist.