A counter question to the OP is what will Harry bring with him that MM Harry won't have? I think the most obvious answer is soulfire. If MM Harry hasn't touched one of the tainted denarius coins he won't have access to hellfire, but even if that is the case I'm sure he'll have some other nasty abilities he can draw upon.
I don't think we can be certain that Harry will be able to call upon Winter to push through obstacles and ignore pain. It's possible that whoever is Winter Knight in the MM world will keep all their Winter Mojo and Harry will lose his. Perhaps the power of Winter will get divided between himself and the MM Winter Knight or perhaps while Harry is in the MM world there will be two Winter Knights around.
If the later is the case I imagine Titania wouldn't be too happy about that. Harry might have to dodge attempts by Summer to reduce Winter's seeming 2 to 1 advantage in Knights. If Harry does have his full Winter power I don't think he has to obey Mab because she won't be his Mab. If Harry were to lose his Winter Knight mojo I would have to question whether his back has healed enough for him to be as athletic as he will probably need to be. That is why I think Harry will maintain at least some of the Winter Knight's ability to push through pain and keep going where a normal person would falter. Plus, he will still know Winter spells.
I think regular Harry will be mentally and emotional tougher than MM Harry. You could even say Harry will be more emotionally mature than his MM counterpart. Yes I know, the name Harry Dresden and the words emotional maturity don't usually go together, but in this case I think they do. MM Harry will be suave and appear to be more confidant and assured. Plus, he will have some magical abilities regular Harry will either not have access to or may have access to but ones he hasn't practiced, like necromancy. However, MM Harry's confidence will hide a deep insecurity that at some point in the story; most likely their final confrontation, regular Harry should be able to figure out and play upon.
I thought he'd have his Winter Knight Mantle, but now I'm thinking not. It will be another way of weakening him to fight his alter. However it will give him insight into how the Mantle had been changing him without noticing it. It will help him to have more control when he returns.
Harry Prime | MM Harry
Stable | Volatile, and unpredictable
Cares about people | See's them as tools or objects
Love / compassion | Anger, hatred, ruthless
Embraces creation | Embraces destruction
These personality traits will give you an idea of what fuels his magic I think. I think Prime Harry will be simply outmatched by MM's raw horsepower. He will also have the unique perspective of witnessing just how frightening he can be, something he couldn't have known until facing himself.
Harry has used both Soulfire and Hellfire. I'm wondering if MM Harry will be using something else. Anti-life power, or spirit energy or something like that connected to necromancy.