My shooting style is mixed, some landscape, some portrait sometimes concerts / shows.
For landscape work: The greater dynamic range and wider field of view with the same lenses of the 6D will work to your advantage. Of course, that is assuming your lenses are compatible with full frame cameras. Otherwise you will need to replace any APS-C only lenses, such as Canon EF-S, Sigma DC, or Tamron, Di II lenses to use them with a FF camera.
For portrait work: The shallower depth of field possible with the FF camera will work to your advantage if you desire such a look in the portraits you take. This would normally be the case for tighter portraits in a setting where the background is not under your total control. If your portrait style is more environmental or full body, then you might not want/need the shallower DoF allowed by the larger sensor.
For concerts/shows: It's more of a mixed bag here, and a lot depends on exactly what type of shows in what type of venues as well as what your access will be in terms of shooting distances. Concert/theatrical work is one of the most challenging forms of photography there is in terms of pushing both the equipment's capability and the photographer's skill and knowledge to the absolute edge of their limits.
Photography is the art of capturing light. Most concerts don't offer
much light to capture and what light there is to capture is changing
rapidly and the subjects are usually very animated. So the traditional
solution to not much light (longer shutter speed using a tripod to
hold the camera still) doesn't work because nobody on stage stands
still for 10-15 seconds while you take a picture. The traditional
solution to capturing motion (faster shutter speeds) doesn't usually
work because there isn't enough light to capture a good image on a
small sensor with a narrow aperture. In the end you have to balance
the two as best you can AND use gear that allows you to capture as
much of the scarce light that is present in the scene in as fast a
time as possible. That means fast lenses (wide apertures), larger
sensors, and cameras that are highly responsive (fast handling).
The 80D has a more sophisticated and configurable autofocus system (but not necessarily more accurate or consistent).¹ The 6D will produce better image quality in low light. Especially if you are shooting under limited spectrum LED lighting that has become popular in smaller clubs, the larger pixels of the 6D will give you more headroom to expose one color channel that may be much brighter than the other two without blowing out that one channel. For more in this regard, please see this answer to Blown out blue/red light making photos look out of focus.
For arena type shows staged with lots of theatrical lighting the 80D will give you more "reach" from the same lenses if you can't get as close as you'd like. For smaller venues which tend to also be darker there is no substitute for the better low light ability of a full frame camera. If I'm shooting a show well lit with theatrical type lighting I'll usually go with a FF camera and a normal (24-70mm or 24-105mm) constant aperture zoom as well as a 7D Mark II with a 70-200mm f/2.8. If I'm in a dark club it's FF cameras only with fast primes, usually a 50mm f/1.4 and a 100mm or 135mm f/2. That and every trick I've ever learned to get non-blurry action shots at slower shutter speeds.
Beyond all of that, you need to consider how many of your current lenses are compatible with a full frame camera. Any Canon EF-S, Sigma DC, or Tamron Di II lenses you have are APS-C only and don't project a light circle large enough to cover a full frame sensor. Even your lenses that you can use with a FF sensor will give a wider field of view on the FF 6D than on your current 600D or the 80D.
¹ The Canon EOS 6D Mark II, introduced in 2017, has basically the same AF system as the EOS 80D. As of 1/24/2019, the 80D sells for $999 from authorized Canon dealers in the U.S. and the 6D Mark II sells for $1,299 from the same dealers.