Full Frame Light Question

Having a larger, full frame sensor means lower pixel density, which means you get less noise. Does it also mean that, being bigger, it physically collects more light if apeture and shutter speed are the same? If so, does this mean in real world conditions that in a given situation you can use, say, a quicker shutter speed than an APS-C sensor and achieve the same exposure, or a larger apeture and achieve the same effect?

Thanks