Yes, it depends. But without putting too fine a point on it--for the moment
--probably mild New Mexico. Better flavor, similar color. Ancho can be a good choice. It has more flavor than paprika but is much darker. In quantity or in some applcations it might be too dark.
Putting a finer point on it: Cayenne is often an ingredient I ditch in favor of a chile with heat and flavor, something cayenne lacks, so I either use a hotter chile as the chile component or combine two or three, one or two of those, at least, for heat and flavor.
For longer cooked beef I tend to go with darker chilies with deep flavor, especially guajillo and negro pasillas but will also include ancho and/or hot NM sometimes. Aleppo often figures in.
I use similar combinations for butt. For shorter, high heat pork items (tenderloin, babybacks) I often go with a lighter color chile like aji amarillos as the primary, and boost with Aleppo or a NM. This is a good approach, imo, for light-skinned poultry as well, and often for fish and seafood. (Try shrimp fra diavolo using straight Aleppo instead of the typical crushed red. World of difference.)
Also, when working off of typically written rub recipes (where the paprika appears at or near the top) try forgoing the paprika--or whatever you're planning on subbing--and make the rub with the other ingredients first. Blend well, then add the chile component(s) a T or two at a time, till you achieve the flavor/color that seems right to you.