The larger the sensor size the more real estate there are for more and/or larger pixels. Since you've constrained the number of pixels to the same as whatever is on the APS-C sensor, then the pixels will be larger on the larger, full-frame sensor. Larger sensors have less noise and more dynamic range than smaller pixels regardless of the actual sensor size.
The other difference is the focal length required for any given angle of view. An APS-C camera using a 50mm will produce the same angle of view as a full-frame camera using an 80mm. However, since the focal length is shorter with the APS-C camera, the depth of field will be larger, too. So if you want the super-shallow depth-of-field look, then the best option would be to get the largest sensor that you can afford. For most people this is a full-frame/35mm format sensor. The ultimate would be to get a medium format camera like a Pentax 645Z, Fujifilm GFX, Hasselblad, etc... These medium-format cameras have larger sensors which use longer focal lengths than even a full frame camera. Thus they will produce a shallower depth of field. And since they have larger sensors with larger pixels, they produce less noise and better dynamic range.
Now, how much of that increase will you notice in your shots and whether or not that difference is worth the added expense is up to the individual.