If you have a PD or non-standard hardware, then a remap is the way to go.
I would say if you have a standard VE 90 or 110bhp TDI, it is most economical to get a good tuning box.
There are vast differences in the quality and type of tuning boxes out there, so preferably get a used box from a known good manufacturer, Van Aaken, Steinbauer, Dragon, Alpin, PSI etc etc, the list is endless. Even VW make a tuning box (havent tried it).
If you know where to look you can get a very good box for under £30.
Preferably get one that has two or three adjustment pot's, so that you can alter fueling/power, the revs power comes in, and how the power curve ramps up. These usually have wires for connecting to the accelerator pot, and MAF aswell as the fuel pump.
One of the boxes which also splices into the MAP connector to allow adjustable boost pressure is a good idea, but only if you have a boost gauge fitted.
If your car has uprated hardware (clutch, turbo, injectors) a remap is much more sensible, and will release more power in a safer fashion.
Many people scorn tuning boxes (as did I, once) and in some respects they are right, but mainly tuning companies want to sell a 'remap' over a hardware mod, as it is more proffitable, 'mostly' on a modern car there is no physical product, just information, which can be sold over and again.
A tuning box will not melt your pistons, or damage your engine any more than a 'remap' will. Any increase in engine output will shorten the lifespan of the engine.
I guess what i am basically saying is that altering the parameters & values in the ECU can be done by remapping or on the fly by modifying ECU/sensor input/outputs in real time. A remap is just a more elegant solution.
To all intents and purposes there is no difference, the result is the same. It is simply a matter of how complicated you want the electronics to be.