Long-time DC5 owner and fairly newish MKIII owner here. I'd say both "can" be made to sound similar to one another given some knob tweaking. The MKIII has a better, rounder clean tone with nice bite on top, and much more headroom than a DC5, if the latter is important to you. That said, the DC5 has great clean and a slightly better "touch of hair" tone on the clean channel for my taste.
With OD, the MKIII is rude, aggressive, downright in your face ...in a good way of course. The DC5 "can" get that way, but I think is more naturally a very fluid OD-tone to it, and verrry creamy with a couple of simple tube swaps ...very pleasing in its own right!
Given that I own both, I can't help you since I love them both ...the DC5 for all that it's done for me in the past and for the seriously good "core" tone that offers great versatility. Then there's my new love, the MKIII, that is brighter, sharper, more aggressive ...for when I want that vibe.
If I *had* to own only one, I'd choose the DC5 for its versatility: 2 completely separate gains and tone stacks and an assignable GEQ offer LOTS of tonal ground on tap. But if channel switching in a live setting is not an issue for you, then I just may go for the awesome MKIII muscle --which you should definitely get with a GEQ for real versatility. Hope this helps you somewhat.
Edward