[mike]

Why would there be more noise if I'm not changing the ISO, and the sensor is the same? ?

At 100% on the computer screen, they would be comparable. You are looking at the same pixels. Only minor differences if the exposure times were slightly different (If the shots were setup for the same FOV and DOF, the aperture would be bigger on the aps-c - more light coming in - and the exposure shorter). For practical purposes we can say they're the same.

The difference comes when these images are printed. There are less pixels/cm^2 on the actual print on the aps-c. Ergo, the pixels are "blown up" more. There are less pixels/cm^2 to cover a given area for aps-c. It's analogous to viewing the image at 50% vs 100% on the flat screen. At 50% you might not see any noise. Whereas at 100% you can. The aps-c print is closer to being that 100% than the FF for any given print size. Again, I'm assuming the same FOV to compare apples to apples.

It's similar to how a high ISO images looks just great on Instagram or a web page where the image is only viewed within 1200x1200 pixels or less area on the flat screen, but if you printed the image to an A3 size it would look horribly noisy.

YMMV. For your purpose you may never notice the difference.