1600 TP with PICT 3 carb.
Currently has a 009 distributor.
Which is best, a mechanical or vacuum advance distributor, and why?

I have a 1776 TP With Pict 3 carb. I am running a Vac distributor. It gives me smooth driving, good economy and bit more low end torque. I am running a 100 type cam

If I was after power and revs and some harder and more hectic driving, running a 110 cam, I would go the Mech 009.

They are different. It is not about what is better.

An SVDA is the best option. Single Vacuum Dual Advance. Basically a combined vac and mech dizzy. The vac does most of the low revs advancing and the mech does the higher revs. The ZV/PAU 4R 8, IIRC, were such a dizzy and used on the 36HP engines. Then afterwards the majority of VW dizzys were vac only. From memory only the early 1200 powered Kombi's used a mech only dizzy but theis was due to it needing to be driven with your foot to the floor to make the thing move.

It's mostly personal opinion and many swear that their 009 drive just fine but I've driven a fair few with 009 and some kind of vac dizzy. The vac dizzy always pulls better from low revs and drives smoother. This stands to reason as VAcuum is almost instantaneous and centrifugal advance (mech) is like a turbo. There's always lag as the weights need time to over come their inertia and spin up to the engine speed to advance the dizzy.

LIKE Psimitar says and esp for a stock or stockish engine ESPECIALLY WITHA A 34 PICT, SVDA with out a doubt
or run the the 009 and suffer from the flat spot and poor economy.

Many think the 009 is a power secret they are confused and misled. 009s bog ( flat spot) down and struggle until the revs are up the sudden release of power is no more than the engine finally running right and not stumbling waiting for the advance to catch up. People think that this is some magical extra grunt in reality its the same power but it seems to surge as the engine has bogged down due to a low base as there has been no vac advance. This is what Psimitar is talking about the lag.
With a a SVDA (single Vac dual advance) the Vac side smooths things out until the mechanical advance locks in this low end drivability is what you want .