I didn't jump directly from those two models but I did try an S20 FE 5G and A13 5G from a Thunderbolt and S4 Mini for a time, so there's a lot of differences there.
There's tons more notifications and many of them insignificant but ALL of them make sound. ALL of them, from 'configuration update' to 'Bixby'. Bixby is another, if your phone has it. I am not sure which A-Series support Bixby Voice but it's Samsung's alternative to Google Assistant. It is quite handy as it can do some things Google Assistant can't do (such as display recent apps, reboot the phone by voice, and automate tasks similar to Tasker) and it also does some things WORSE than Assistant as well (pretty much in the same vein as S-Voice when asking it for online facts)
You'll probably spend a good few days of muting individual services and turning off notifications unless you actually like the cacophony of sounds going off every hour or so. Get used to hearing whatever default sound they use for notifications (some variant of Over the Horizon). until you turn those off. It'll be like pulling weeds--soon as you think you got 'em all, another shows up.
The software is called One UI (the S6 used TouchWiz at first then got its last update with the 'Samsung Experience UX' aka Android 8.0) and is currently on version 5.1, and it's a lot different from what you're probably used to. It's quite flat in its default look but there is a theme store that offers alternatives, icon packs, complete themes and so on.
There's also USB-C, which I don't recall if the S6 had it or not. It's been awhile (I charged mine wirelessly) but USB-C can get weird. Often times the phone wouldn't charge unless I plugged in the cable twice and sometimes I'd wake up and see it showing '18% charging, unknown time remaining' when I put it on charge before bed at 65%. USB-C has this bad habit of making the phone think it's charging when it's still draining. Never did figure that one out. I had to either use my S20 FE on a wireless charger stand to work around that or use my A13 5G on the cable with the power off since it never finished charging with it on.
They don't include an offline music player out of the box. The Samsung Music app is available on Play Store or Galaxy Store whichever is your preference. If you're a streamer you can install whatever service you want as well.
The Samsung Internet browser on these modern phones also supports extensions like adblock, and their messaging app supports wallpaper and custom themes (three dot menu-->customize chat room) and supports RCS chat features if you use those.
I hope you like gestures since Samsung sees those as the future. For now they come out the box with on-screen navigation buttons, but I wouldn't be one bit surprised if later an update forces the gestures so I'd get used to them now while you still have the choice. I couldn't get used to them since my dexterity is awful and need buttons, but your mileage may vary.
Battery life is average, not perfect but you should get a day or two out of it. My A13 did better than my S20 FE did, and could go two-three days of charge.
Overall those are the major differences I dealt with.