I've just found the Wrangler weapon for the Engineer, and, after reading its description, I'm not sure in what situation is could possibly be better than letting the Sentry automatically do its thing.
The disadvantages of using the Wrangler are:
It makes me an easy target for spies. Whilst I'm fiddling about trying to aim at people, a Spy can easily back-stab me, and then have plenty of time to sap my Sentry (as it takes a while for the Sentry to go back to automatic mode).
Sentries have much better "reaction times" and can detect and shoot enemies much more quickly than I can.
A Wrangler-operated Sentry gun has a laser pointer (and a blue/red bubble surrounding it), which gives away my location (and exposes the fact that I'm too busy trying to shoot individual targets to notice someone ambushing me).
Sentries never miss their targets, whereas I am likely to miss, since I am human.
The Wrangler takes up a spot in my inventory, replacing my Pistol.
Sentries, when operating automatically, cannot accidentally shoot their owners, whereas, using the Wrangler, I'm highly likely to accidentally shoot myself if a Pyro/Scout gets too close to me without warning.
So, in light of all these disadvantages, I have to ask: what are the advantages of using the Wrangler (i.e. are there any situations in which it would be better to manually control a Sentry, than to let it shoot down enemies automatically)?