Hey elford i would sell my entire kit to go the ff route. What faster glass are you referring to?
He meant keeping your current camera, not buying a full-frame camera, but instead buying "faster" ( = larger aperture) lenses so you can shoot in lower light anyways. Those are probably prime lenses because zoom lenses don't go lower than f/2.8, and you already have that as your fastest lens. So, he will be referring to primes such as the 24mm f/1.4L, 35mm f/1.4L and the 50mm f/1.4 (mentioned earlier) or f/1.2L.
Compared to your f/2.8 lens, f/1.4 is exactly 2 stops "faster" (it lets in 2x2 = 4 times as much light).
If the 5D mk3 is 1 stop better in noise performance than the 5D mk2, then the 5D mk3 would give you 3 stops increased light performance compared to your 60D at similar apertures. But because the kit lens is one stop slower (f/4 instead of f/2.8, a 1 stop difference), the effective speed difference is also 2 stops, or 4x as much effective light intake.
Advantages to the "glass" route: you'll have glass which will last you longer than a camera body. Your lenses will deliver top notch sharpness. You'll learn to zoom with your feet giving you more input into the photography process. It may be cheaper than the body depending on the lenses you would pick (I'm not going to look up the prices though).
Disadvantages to getting "glass": you'll need to carry around several lenses for different focal lengths, and switch them when you want to use a different focal length. Low light shooting forces you to use the aperture wide-open, which gives a very shallow depth of field, making it hard to keep the subject in focus (but producing very nice blurred-background pictures when you hit).
Advantages to the "ff body" route: higher usable ISO gives you low light abilities even with slower ( = smaller aperture) lenses, which makes it easier to be versatile (because you can use zoom lenses instead of prime lenses) and easier to keep the subjects in focus (more depth of field due to smaller apertures).
Disadvantages: a body will last you a good while but not forever. It's expensive, and re-sale value is probably a bit less than for decent lenses.