I already have Canon EOS 600D and I am thinking of upgrading Camera now after 5 years but I am confused over Canon 6D which is a bit costlier for me and the newer Canon EOS 80D which is in my budget. I want to know do I get significant improvement Image Quality wise if I upgrade to Full Frame now.
PS: I do not have budget for 5D series yet.
Either way, image quality will be improved dramatically. The 600D and 6D are roughly of the same generation but, since the 6D is full-frame, it will deliver much better image quality. While the 80D is APS-C, it is considerably newer which has seen much improvement in performance simply since technology has evolved.
Between the 6D and 80D, the difference in image-quality is relatively small. Image noise at low ISO is quite similar but the full-frame still shows a significant advantage at high ISO. The dynamic-range from the 80D is actually slightly better than that of the 6D. If you shoot often in low-light, than a 6D will give you the most improvement. If you shoot mostly at low ISO, either in bright light or with a tripod, there will be a much less noticeable difference.
Since you are considering stretching your budget for a 6D, remember that some or all of your lenses may no longer work. The two nice things about the 80D is that, you get to keep all your lenses without change in field-of-view and, unlike both other cameras, it has a 100% coverage viewfinder, so you can see everything that will come out in your photos. This has nothing to do with image-quality per-se but it will improve your photography as you will not have to worry about what may appear in the frame. Plus, much fewer pictures will need to be cropped.
Whether this is an improvement or not depends on your shooting style. But it is certainly a significant change.
The change of sensor size means that your APS lenses will become useless. Your full frame lenses will behave differently - they will have wider angle of view and you may even notice that the character of their rendering somewhat changed. If you keep only your existing FF lenses, you will probably notice that you need to readjust your previsualization - the lenses will show wider field of view the before.
And this is not just lenses. Different cameras have different shooting speed, different AF implementations, different noise level at different ISO settings. It looks like the 80D may be a better camera for shooting sport and 6D is better for low light, landscape and portrait. With the 6D it is more likely that you will want to do some upgrades to lenses, so expect additional cost...
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.
I have been using the 6D for almost 4 years (I think), shortly after it was released. I'm very happy with the 6D and the quality images from the full frame sensor. But you certainly don't need a full frame sensor to be able to get quality images. You can get great images with the 80D too.
Do you have lenses that will be compatible with a full frame sensor? Canon's EF-S lenses will not work with the 6D, so if you would need to upgrade your lenses too it will be much more costly.
If you are looking to make money from your photography you can probably justify the cost of going to full frame since it will give the potential for higher quality images. If it is mostly a hobby and you would also need to upgrade your lenses, I would stick with an APS-C sensor.
