The "problem" with full frame cameras is that they require full frame-capable lenses. Those are necessarily bigger, heavier and more expensive than lenses for APS-C cameras.
APS-C cameras, on the other hand, can use both crop only AND full frame lenses.
Compare a wide angle crop only zoom such as EF-S 10-18mm ($279, 67mm filters, 270 grams) or EF-S 10-22mm ($599) on APS-C with full frame EF 16-35mm f/4L ($999) or EF 16-35mm f/2.8L III ($1899, 82mm filters, 790 grams).
For sports/wildlife you'll probably want a telephoto much of the time. Even comparing EF lenses to EF lenses, you'll need much larger, heavier and more expensive ones for full frame than for APS-C. For example, if you use a 55-250mm now, you'll want a 100-400mm with full frame. Or if you use a 300mm, you'll want a lens that reaches 500mm. Or if you use a 400mm, you'll need 600mm to be able to frame distant subjects the same way as you do now with APS-C cameras. For example, compare an EF-S 55-250mm ($299, 58mm filters, 375 grams) with an EF 100-400mm II ($1899, 77mm filters, 1640 grams). Or compare an EF 300mm f/4L IS USM ($1349, 77mm filters, 1190 grams) with an EF 500mm f/4L IS USM II ($8999, 52mm drop in filters/130mm diameter front element, 3180 grams). Oh, and you'll probably want a sturdy tripod for that 500mm lens, too... so figure another $1000 to $1500 for that (not to mention it's size and weight).
Will full frame make "better images"? Yes and no. While viewing the images HUGE on your computer monitor OR if you make really big prints from them, you'll see the superiority of image made with a full frame camera. By "huge" I mean what a lot of people do automatically, looking at their images "at 100%". On most modern monitors, with a 24MP camera that's like making a 40" by 60" print (five FEET wide)... and then viewing it from 18 or 20" away. Ridiculous! Few people will even print half that size. BUT, for most peoples' real world uses, a modern APS-C camera will fulfill their needs just as well. At more normal sizes 13x19 or 12x18 or smaller... and especially for much lower resolution and far smaller Internet sizes... you won't see much or any difference between images made with APS-C or full frame.
So, it depends a lot upon what you plan to DO with your images, whether it makes sense to spend the extra for a full frame camera AND lenses... as well as deal with the larger sizes and extra weight.
Both cameras have faster frame rate and more advanced AF systems that would be better for active subjects than your current camera. Both also offer a moderate upgrade in resolution from your current 18MP camera.
80D is 24MP and about 4 years newer than 5DIII. 80D introduced a new 45-point AF system, all cross type, with up to 27 points able to focus at f/8 and rated to -3EV.
5DIII is 22MP and wit's AF was much improved over the previous 5D series model. It uses a 61-point AF system, 41 of which are cross type, f/8 at the center point only, rated to be -2EV capable.
(Note: f/8 focusing allows a lot more lens/teleconverter combos to be used. Many cameras that are "f/5.6 limited" simply are unable to focus them.)
Also keep in mind that the older 5D Mark III was superseded in 2016 and Canon will at some point no longer support it with spare parts and factory repair services. The 80D is still in production, so there's good possibility will remain serviceable further into the future.
Local camera shop is full of B.S. With newer models, I don't think twice about shooting ISO 3200... twice the limit your local store is telling you. Below is a high ISO, low light test shot I did with one of my 7D Mark IIs at ISO 16000 (not a typo.... sixteen
thousand, not 1600):
Sure, there's some "noise" in the above image... but IMO it's very well controlled for such high ISO. I'd make an 8x10 or 8x12 print from that shot, though probably not any larger.
I've sold a lot of images made at ISO 6400, 8000, 12800 and even a few at 16000. I shoot RAW and do some extra post-processing and careful noise reduction on those very high ISO shots.... Above image was NOT as extensively post-processed. It was shot RAW by only a single 60 watt bulb and a small window, both about 8 or 10 feet from the subject. To see the "worst case", I deliberately only applied default level of NR to the image in Lightroom (none is applied in-camera when shot RAW). The only adjustment to that image was a slight increase in contrast. That actually should have made existing noise more apparent.
I recommend checking out Bryan Carnathan's reviews at The-Digital-Picture.com He is very informative, specializes in Canon gear (though he is knowledgeable about other brands, too) and does thorough testing. I've found his reviews to be spot on, for any gear that I use.
The "problem" with full frame cameras is... (