Full frame is a mixed bag, in terms of pros and cons, unless you have a budget at least around $4-5k (including the body).
The 6D/5D3 sensor is substantially better than crop sensors. It will give you slightly cleaner images at low ISO, and much cleaner images at moderate to high ISO. Obviously, that's a positive.
Unfortunately, image quality is a function of both the sensor and the lens. Which factor is the bottleneck can vary depending on the situation, or even across different parts of the image. I say "unfortunately" because, in general, lenses that are sharp across the larger image circle of the full frame sensors are substantially more expensive.
On crop, lenses like the EF-S 10-22mm and EF-S 17-55mm IS can be had for $600-800 apiece, and will give you awesome sharpness and image quality throughout the frame. In fact, some respectable third-party alternatives to those lenses go for under $500! In general, it's much harder (and more expensive) to achieve that on full frame. Similarly-priced lenses like the 17-40L, 24-105L, etc. tend to have soft corners and other more noticeable flaws on full frame. For example, the 10-22mm and 17-40L are about the same price and serve roughly the same function on crop and FF, respectively. But in my experience, the 10-22mm was a lot sharper in the corners on crop than my 17-40L is on FF.
To get corner-to-corner sharpness on FF, you often need to look at primes or very expensive zooms like the 24-70L II. Some primes are relatively affordable, but you need several of them to cover the range that one lens like the EF-S 17-55 IS covers perfectly well on crop. Either way, you're looking at a much higher budget for good lenses on FF.
You really have to weigh the pros and cons in the context of your budget. If your budget is just high enough for a 6D, but not enough for several top-end L lenses, are you sure the lower noise is worth the potential tradeoff of softer corners? That was actually the decision I had to make recently, and I decided it was worth moving to the 6D from crop, with the hopes of acquiring better glass over time. But based on my experience thus far, for some applications, one could easily argue I spent a lot of money to get results whose advantages (cleaner files) are offset by other disadvantages (softer corners). The Samyang 14mm prime is a glaring exception, offering corner-to-corner sharpness for bargain-bin pricing ($350). But it's a very specialized lens, and one whose value cannot be replicated at most other focal lengths.