Hmmm. That's a tough one really. At first glance, Vanguard should look to be hands down better, with Adrenaline Burst and Medium armor. But as alluded to above, when you build them out it isn't so simple.
The main problem will be figuring out how to invest the points and what to skimp on when you add Singularity to the Vanguard, or even Shotguns to the Adept. But overall a shortage of points is a little more acute on the Vanguard. Personally, I wouldn't skimp on Marksman / Pistols since their dps is much higher than with shotguns. In fact, IIRC on the last Adept I played, which had shotguns, I think I didn't actually invest points in Shotguns until around mid to late game. And by the end, I don't think I had more than basic Carnage. You can get away with this because as long as you have the talent, you can zoom the weapon.
As red mentioned, there is indeed a glitch with using Carnage and Basic or Advanced Barrier. If that barrier is active, Carnage will drop them. That was part of the reason I delayed getting Carnage in the above adept run. On a standard Adept or even some Vanguard games I don't end up going past Advanced Barrier, but it is a good idea if you like to use Carnage.
Anyway both are good though. Here is what I would probably shoot for with the builds:
Level 50 Adept
Barrier 12 (Master)
Basic Armor 5 (Shield Boost, Unlock Pistols)
Pistols 12 (Master Marksman)
Throw 6 (Basic, Unlock Lift)
Lift 12 (Master)
Warp 7 (Advanced, Unlock Singularity)
Singularity 12 (Master)
Shotguns 7 (Basic Carnage)
Bastion 9 (Barrier Specialization)
Spectre Training 4 (Unity)
Intimidate 10 (4 freebies)
And I would level it more or less similarly to the route I wrote about here. I personally don't care for Stasis and don't find I need yet another CC power, but if you like it, drop one point out of Shotguns to pick it up. Speaking of which, you will hit Advanced Carnage at 51 or 52 depending on if you invest in Stasis. That also has 6 points in Intimidate invested so you can get all persuade options, but if that isn't important to you, you can skimp on those.
Level 50 Vanguard
Assault Training 8 (Advanced Adrenaline Burst, Unlock Tactical Armor)
Barrier 12 (Master)
Tactical Armor 7 (Shield Boost, Unlock Medium Armor)
Pistols 12 (Master Marksman)
Shotguns 4 (Carnage)
Throw 7 (Basic, Unlock Lift)
Lift 12 (Master)
Warp 4 (Basic, Unlock Barrier)
Singularity 7 (Advanced)
Shock Trooper 9 (Barrier Specialization)
Spectre Training 4 (Unity)
Intimidate 10 (4 freebies)
Taking Adrenaline Burst Specialization can be done at this level, at the expense of Advanced Adrenaline Burst. Either way it should have a 90s cooldown, the choice is between passive bonuses in Assault or the Shock Trooper talents. More damage in Assault, protection in Shock Trooper.
You probably don't need a long duration Singularity when you have Adrenaline Burst, so I have not invested past Advanced.
If you just wanted to use Shotguns to use them, I would probably drop points from there, drop Barrier to Advanced, and increase Warp and Throw with those points.