Member management is somewhat different between the two apps, although you still can’t limit which devices a secondary member can turn on and off.
However, in the V3 app, secondary members are not allowed to delete devices or locations. Also, you can create a QR code and use that to invite them.
Everything is quite different for classic
The official “SmartLights” feature is somewhat different between the two apps, as well as between iOS and android.
The following page discusses all of these. There are tabs in the middle of the page which will let you switch between the “SmartThings” (new V3 app) instructions and the classic V2 instructions, so check to make sure that you are on the one that you want.
https://support.smartthings.com/hc/en-us/articles/205380404-Smart-Lights
The official “Speaker Companion” feature is installed differently in the two apps, but otherwise looks about the same. There are also some differences between iOS and android.
The following discusses all four of these. There are tabs in the middle of the page to swap between V3 and Classic, so make sure you were looking at the one you want.
https://support.smartthings.com/hc/en-us/articles/115002004243-Speaker-Companion
To the best of my knowledge, ActionTiles can be installed (via the ActionTiles website https://app.ActionTiles.com) and used regardless of which SmartThings App is used.
As of this writing ActionTiles cannot trigger the new App’s “automations” or “scenes”, nor control the alarm (SHM replacement) features.
As of this writing, if you have an android phone, the V3 app can connect some Bluetooth devices, but know that they’re being connected to your phone not to your hub. For that reason you will not be able to include them in automations.
Classic will not see these devices at all unless you are using some custom code specific to that device and some kind of “man in the middle “server, but in that case you will be able to include them in automations. (if you have an android device, you may also be able to use classic plus Tasker plus sharptools.)
So V3 has a much simpler set up but classic can do more with these kinds of devices.
As of this writing, Smart Home Monitor in STSC is a totally separate app than in Classic. It uses the new API and has different features Specifically:
To clarify, Keen vents work with STSC, but there is no way to add them. At least they aren’t listed as a supported device. So you have to add them via Classic.
1.6.05 came out on iOS yesterday and supports iPads. Haven’t tried it myself, yet.
Supposedly the ADT panel works with STSC now, but i have no way to verify that.
As of this writing, Support still is telling people With the ADT/smartthings security panel Model to use classic for now. They are beginning to port over the features, but that work is not yet done.
New ADT customers should use the SmartThings Classic application. We are still working to ensure ADT features are fully supported in the new SmartThings app.
https://support.smartthings.com/hc/en-us/articles/360000114006-New-SmartThings-app-FAQ#NewHub
The current version of STSC does not support landscape mode for iPads.
Supported as of version 1.6.39.
You cannot initiate a Z-Wave network repair from STSC.
In Classic, you cannot enable Zigbee OTA device updates.
In Classic, you can invite users to your location via email.
In STSC, you can invite users to your location via Samsung account or QR code but not email address.
In STSC there is a device called “Smart things hub”, which can be assigned a room. This device is not present in the Classic App. If by accident you delete this device, instead of removing it from a room per se, it causes a partial hub deletion and screw your entire setup up, causing you to have to re add every device. Devices will still show in the Classic App, but no zIgbee functions work, and zwave will be shoddy and not work until you re add everything.
One thing I have noticed in STSC is that the smart home monitor does not update alarm status when changed in classic or on Action tiles. If you had the status as armed away in STSC then disarm it on action tiles stsc would still alert you that there is a break-in.
That’s because the new apps SHM is totally separate from the Classic version. It’s a whole different app using the new API instead of groovy.
@markdewitt2 … He’s correct.
What I don’t know: Is the new Solution also exactly called “Smart Home Monitor”?
You’d think I’d know by now, but to be 100% honest, I’m avoiding the new SmartThings App as if it were ebola. Head buried in sand here, but I … don’t like this ambiguous situation at all. 
.
Yup, same name and everything, but doesn’t have leak, smoke, and custom (yet?), just security.
Update 8/11/18: leak and smoke added to SHM in STSC.
Edit: new app now calls it SmartThings Home Monitor
As of this writing, STSC does not offer any way to automate the unlocking of a lock. Just the locking of it. You can open the mobile app and see the lock listed and unlock it there. But you cannot set up to unlock it when you arrive home, or at a certain time of day, or when a specific sensor activates. You can do all of those and more with the classic app. It is unknown whether this capability will be added to STSC or not.