I shall add my opinions to Piccolo's excellent summary:
The Dualogic selector has the gear up/gear down lever as the primary function. 'Auto' (not called 'D') mode engagement is a secondary function. It is not the same as a standard automatic selector, which has D (Drive) as the primary function and gear up/down as secondary functions.
This difference pretty much summarises the whole Dualogic experience. You will get better results treating it as a manual - tell it when to change up gears, and it works efficiently. I get smooth results better than I would with a regular manual - no chance of stalling, no need to change down when coming to a stop (it always does that for you). Switch off the engine and it's parked in gear, which is nice too.
With a 1.4 engine, it's worth pointing out that there are two modes - one mode gives smoother, slower gearchanges by backing off the engine power, while the Sport mode gives a faster but more noticeable gearchange as the engine power is not reduced as much. I drive in Sport mode all the time, as the engine feels much more responsive this way. In New Zealand, the Pop model came with a 1.4 engine until recently - with a 1.2, there won't be the benefit of the Sport button and so I can understand what Piccolo means about the 'leisurely' gearchanges.
'Auto' mode is best thought of as a novelty and is unlikely to satisfy the driver who wants a standard automatic experience of never having to select a gear. They will instead feel the Dualogic shifting noticeably between gears as it tries to guess what's best for the conditions.
However, it has to be said that conventional automatic transmissions attached to small engines are often caught out in difficult conditions such as hills, corners, speeding up and slowing down - where they are likely to feel hesitant. Manually overriding the automatic may help, but often doesn't (a delay in response or a large gap between ratios being typical problems). There is also a performance and fuel economy penalty with a standard automatic transmission due to the 'slip' introduced for smoothness. One type known as CVT (Continuously Variable Transmission) takes some time to change ratios (moving the belt across pulleys takes time, a bit like bicycle gears where the chain moves across sprockets). Therefore, CVTs in particular feel quite lethargic when moving off and there is nothing you can do about it.
I really like my Dualogic - but here's my unbiased advice: take both the Dualogic and other small automatic cars for a test drive - make sure your test includes slow-speed city driving and medium-speed B-road driving with corners and hills. Motorway driving is less important to test, as any transmission will simply be in its highest gear, though you might consider the pickup during overtaking to be important (does it change down correctly to ensure safe overtaking?)
I'm inclined to agree with Portland Bill on this one; once you've driven both, you'll know quickly which one suits your needs. One will seem to be faithfully in tune with your driving, while the other will be working against you. You'll decide which the Dualogic is
Also, when reading about the FIAT 500, remember that models sold in the USA have a conventional automatic transmission not available in Europe.
-Alex