Modern autos are generally faster as they can change faster than a human.
depends on the Auto, the older slush boxes were all much slower than a manual - New dual clutch and ZF Autos are far faster.
Even with the faster autos they will never truly replicate that accurate and reassuring connection to the engine that a good manual box provides - and I emphasise a good manual as there are good and bad ones. I have been lucky enough to drive Honda and more recently Subaru manuals - the Subaru 6MT is very good. On a B road there is only 1 winner as to which is better.
So its not a straightforward answer... you will need to drive both and make your own mind up, if you are stuck in traffic a lot it makes sense to get an auto - i would have sold my Subaru as i was in that position but quit my job and moved house..so i could keep the manual.
that's dedication.