Hi adam1991,
The sheets would normally only have a small effect on the mattress and if the mattress protector has some stretch to it then it would also have a relatively small effect as well but a thicker mattress pad can have a more significant effect on the feel and performance of a mattress which can be helpful or detrimental depending on the person. Depending on the specifics of the pad … it can provide some additional “surface” softness but at the same time it can also reduce the ability of foam materials underneath it to contour to the shape of your body and relieve pressure which can also make it feel firmer as well. You can see an example of what I mean from one of the members here that was sleeping on the Tuft & Needle with a mattress pad in post #231 here and this topic.
It may be an idea to remove the mattress pad to get a better idea of how both mattresses feel and perform without it and you can also add it back again if you feel that it’s an improvement on either one or both of your mattresses.
Phoenix