The S&W 625 seems to be one of those handguns people always wish they had bought. It doesn't get as much press and the favorable comments the L-Frames do but, I have had more then my share of customers and friends regret either selling or not getting a 625 in the first place. I will note, that for those people wanting more power (like 357 Magnum levels), 45 Super is an option for these stout handguns. Full Moon clips also offer reloading speed advantages for people shooting bowling pins and pepper poppers. For a "learning" pistol, I think the 625 is a good choice. Total weight, balance, recoil and, muzzle blast are all within reasonable levels.
In the 357 L-Frames, my personal pick of the litter is the 686 Power Port model. This is one tame shooter - even with the heaviest loads. This is a very accurate pistol in my hands as well. You also have the option of shooting real "wimp" loads in 38 Special when you do your initial learning but, I like something a little more realistic like lightweight 38 +P's. Muzzle blast with hot ammo in these guns can be a negative factor in the learning process as well. A 6 inch barrel goes a long way to helping tame the blast but, there's not doubt it's still there.
For general use fun shooting, general learning and, a secondary self defense role, I think I would give the nod to the 625. I like big bullets moving at medium velocities for general use. Muzzle blast with stout 357 Magnums and the recoil generated get old after 300 rounds or more (when I shoot, I shoot a lot, usually 8 hours or more). And, 45's in general will give you a shove and a deep boom instead of a stout slap and a high pitch'ed crack (which does get old at the range).
Either choice you mention is a good one. The intended markets for each is a little different so, the basic question is - what do you really want to do with the pistol? What qualities are you looking for? How much do plan to shoot and what type of shooting will you be doing?
Sid