Enormous difference in velocity is likely. My 3" 65-5 throws 158-gr .38 Special +P factory loads at just under 900 fps. It throws 158-gr .357 Magnum factory loads at around 1200 fps. Handloading can increase both numbers, but the proportion remains about the same.

Now, ask yourself this: what difference does that additional velocity make in a defensive shooting of an aggressor? Maybe not so much. Further, what additional overpenetration dangers are associated with that additional velocity? How much quicker is my follow-up shot with the slower load than with the faster one?

Lots to consider, frankly. I love my .38 Specials, but - all things being equal - I'd rather they were all chambered in .357 Magnum just for the additional versatility that chambering possesses. Which round is better for your purposes depends on what you're doing. When I hike up Bear Canyon with my 65-5, I stoke it with 180-gr hardcast handloads that do 1230 fps from the gun. When I carry it in town, there's no way I'm carrying those . . . .