As someone who owns both cameras, let me share my opinions on both. I really liked the D750, but it wasn't perfect. The flash system (Admittedly I'm using Godox) never seemed to be fully reliable for TTL, and I could never wrestle with FEC the same way I did when I shot Canon
++ Great IQ
++ Fast focus
+ Wireless (3rd party software is the way to go, though)
+ Excellent battery life
+ Feels fast
+ Lots of smart extra bits, like auto-ISO being able to be enabled/disabled independently of the ISO value, and auto ISO disabling when flashes are firing (Smart!)
+ 3D focus is awesome
-- Narrow AF area - not easy to compose action shots properly - this was what attracted me to the A7III or mirrorless in general
- Phase detect focus need calibration for fast lenses with narrow DOF - frustrating to see people's heads out of focus when reviewing shots afterwards.
++ Focusing points out to the edge of the frame!
++ Eye-AF is amazing, and even works (in low burst, 1fps) with adapted glass!
++ Face detect with linked AE is exactly how photo taking should work
+ Great IQ
+ EVF lets you pre-chimp
+ Amazing video capability - Phase detect on sensor!
+ 1080P 120fps is awesome, with AF and sound capture!
+ Good battery life
+ Focus peaking...though
- Focus peaking isn't sensitive enough - I often look at manually focused footage and say "Well that wasn't focused"
-- Flash work is a little frustrating - EVF hides ambient exposure when using flash, and there's another menu setting you need to enable to see what your actual ambient exposure.
-- Switching from different modes feels like rebooting a really fast computer. Doesn't take too long (seconds), but that's ages when you are trying to get a photo.
-- Touchscreen is only useful for selecting the AF point - doesn't work in menus
-- Feels slow - My GH4 from 2014 flips through the menus really quickly, and also lets me use the touchscreen for them!
-- Still can't do much when buffer is clearing. I thought this was over when I heard that you don't see the "Can't operate - writing to card", but this still happens. It's just that you can change settings while writing now. <sigh>
-- Camera plays "Simon says" - if you tell it to write to two cards, and don't have a card in slot #2, it'll say "Can't write, no card in slot 2"! Wow! I don't care, shoot the damn photo. Something else I only noticed about my Nikon when it was taken away. Nikon will shoot RAW+JPG to one card if it's all you have inserted.
-- So many options will disable other options (Proxy recording for Video disables Eye-AF, HDMI out disables Eye-AF, HDR disables RAW shooting, etc.), though it requires so much menu flipping in order to figure out how to turn off the blocking feature that it's more frustrating. At least they tell you where the conflict lies (A little)
-- Menus are terrible, and many of the options shouldn't be necessary. USB connection mode - how about you figure that out automatically?
-- Camera requires a database to use your cards. This kind of freaked me out when I first started using them. Why do you need to set up a database before shooting? Just write the photos. What happens if the database gets corrupted? Several times I've had to rebuild a database, and this makes me nervous, because it feels like I've lost data
- Buttons are a little mushy - no satisfying click
- Cards go in backwards - label is away from you
In general, the A7III feels like a really fast computer that has a lot of stuff to do before you're able to take a photo with it. It happens really quickly, but it's that latency that's really distracting, and frustrating.
The D750 feels like a purpose built camera (Like every other DSLR before it), that has some smarts in the back end that you can turn on without them affecting anything.
EDIT: I should point out what I shoot - most of what I shoot for money is BMX racing, so need fast focus (I really didn't think the A7II would've cut it), and the rest of the time I'm shooting either people in the studio for fun (Godox flashes), or bands playing at bars - low light.
For the BMX, the D750 is great, though does struggle focusing when subjects are backlit. I think the A7III struggles even more with this (Haven't yet taken it to the BMX track)
The main problem I had with the D750 is that I couldn't frame a rider's head at the top of the frame and have the AF system track them (Head was out of the AF area). 3D tracking works really damn well, but not when the subject is outside of the AF area!
One of the other big problems I have with Sony is that there's no used market, so lens prices are very high. Nikon has 3 generations of 70-200 lenses that you can "cheap out" on (VR1 is totally fine for me), but Sony only has the $3200CAD 70-200 FE GM