Hi, in general the EOS 6D is in many ways a better camera than the 60D. Still the 60D is able to take great high quality photos. I've never used either of them but I believe the IQ of the 60D is similar to the one of my 7D. The technical IQ of the 6D is better, the difference could be big or hardly noticeable depending on shooting conditions.

The main differences between full frame sensors and crop sensors is (simplified) that the larger light sensing surface of the FF sensor gives you better "signal/noise ratio". The full frame sensor will simply give you less noisy photos. This is especially showing up when using high ISO settings. At moderate ISO settings the difference isn't significant for typical use, but a large print of an image taken at ISO 3200 will look better if it's taken using a FF camera body.

The other significant difference is that you will have to use a longer focal length on an FF camera compared to the crop camera to get the same framing (the 1.6 crop factor). This could be good or bad depending on the situation. If you want the longest possible reach for telephoto shooting, the crop factor could be seen as an advantage. But on the other hand - if one is a sucker for narrow DOF, the longer focal length required by the FF sensor (for equal framing compared to a crop sensor) is an advantage.

Apart from these main general differences, the 6D is a more recent product which means it beats the 60D on some functional stuff. The main ones are GPS and WiFi.

So, 60D or 6D depends on your needs, shooting style, etc. It also depends on your budget since the 6D costs more than twice as much as the 60D. (And the lenses you need for the FF camera also cost more...)

My general advice regarding gear purchasing/upgrading is don't do it until you've grown out of what you already have - don't let the gear collecting take command over creating photos. Many of us - me included - have a tendency to dream of new gear thinking it will give us better photos. I believe other things than new gear makes more difference, e.g. training, training, reading some books, training and some more training.

Good luck with your choice!