The "AUTO" is a DSG, or direct shift gearbox... it has dual clutch pack to split manual transmission. The controller shifts for you, but it's a manual transmission that shifts better than you could ever do. You can always manually shift it, without having to push a peddle with your left foot.
If you look at almost any super car, Porsche, Ferrari, Lamborghini, Bugatti Veyron and Ford GT.... all have DSG... question?
Which is better? Practically on a VW the manual is slightly lighter and due to final gearing gets a little better MPG for cruise (due to slightly lower RPM for highway speed). This is all goodness, except in ease of driving on a daily commute sipping coffee.... For the DSG... amazing fast smooth shifts. The DSG requires oil and filter every 40K miles you can do for $120 in parts (if you pay for the service it is triple this). The DSG clutch will likely out live the manual clutch by a factor 3. DSG has proven to be reliable. Manuals are also reliable. There is the DMF dual mass flywheel, dampening springs failing (which may be rare, but it's a thing). I think DMF is on both manual and DSG, not sure.
I guess one more issue is if you go for a "TUNE" your VW TDI for more torque, you also have to TUNE the DSG to accept the higher torque, which is limited by the transmission software.... that cost an additional charge to program.
This subject has been beat to death...it is like debating what is the best ice cream flavor, personal choice. Practically DSG is more common. For some it is a sign of manhood to have a manual and envision it makes them great drivers, better than DSG drivers... seriously there is some chest thumping over the issue. Some say MANUAL ONLY ... I say don't care. I have DSG and love it. If I had manual I'd like it as well. Both are fine.