Simply put, Full Frame sensors are bigger.
Crop sensors, or also known as APS-C sized sensors in Canon terms, are 1.6x smaller than Full Frame sensors. (1.5x for Nikon)
There are a few things that will be affected due to this difference in size, with the most prominent one being the field of view.
Basically on a crop sensor, the crop factor applies a crop to the field of view, much like the crop tool in Photoshop. It crops down the edges of your image. This is why you end up with a tighter field of view.
So to calculate that tighter field of view caused by shooting with a crop sensor, all you have to do is take the focal length, for example 50mm, and multiply it by the crop factor. That’s 1.6 for Canon users and 1.5 for Nikon users. So 50mm X 1.6 is 80mm, so that means using a 50mm lens on a crop sensor will give me the equivalent field of view as using an 80mm lens on a full frame.
Full frame cameras also generally have better low-light performance. If the megapixel count is constant, and with the full frame sensor being bigger, that means each individual pixel is going to be bigger on full frame. The individual pixels will be smaller on a crop sensor camera because you have less area to cram all those pixels onto. With bigger pixel sizes, they are able to receive light better and generate less of that digital noise we all dread at high ISO’s.
ISO 12,800 on 5D Mark III (Full Frame)
ISO 12,800 on 70D (crop)