From experience, I can tell you that AllParts bass necks do have a slighlty smaller heel than other Fender bass necks. I had an AllParts jazz bass neck on a mighty mite body and there was maybe a 16th of an inch of play in the neck pocket fit.
Also, I found the truss rod on the AllParts neck to be a bit weak. I was using 50-105 rounds regularly, which aren't particularly heavy, and the neck just wouldn't hold up. I had to constantly tweak it to keep low action. Not an issue if you don't mind the tweaking, but it bothered me after a while.
I replaced my AllParts with a Warmoth neck, and I will never buy a neck from someone else ever again. This thing is solid as could be. I have 55-110 rounds on my bass now in standard tuning and the neck hasn't moved a bit since I set up a month ago.
However, the Warmoth neck is heavier than the AllParts neck by a substantial amount. I never weighed them but I could tell the difference. I attribute that to the additional steel stiffening rods in the Warmoth neck.
Also, there is no guarantee that the AllParts neck will be a tighter fit with your body. MIM necks are decent quality, but for the price being not that far off from a Warmoth neck, I would personally go for the Warmoth neck. Especially if you're picky about fret size and those kinds of details. If you had to, you could rout or have someone rout the AllParts neck pocket out a hair to accommodate the Warmoth neck, if that's even necessary. That would be better to me than a sloppy neck pocket fit.
I will say that the Warmoth neck fits my Mighty Mite jazz body a lot better than the AllParts one did. Nice and tight! Just a tiny gap to one side, which leads me to believe that the MM pocket is too big and that the Warmoth neck is probably as close to Fender as it gets
That is my $.02