The Difference
 

How much shorter (reach wise)are modern handlebars compared to traditional bars? We're not talking super deep drop bars, but "normal" ones. I am looking for a bike with a short top tube and modern bars might help.

As Robbie said, a stem change could be all you need. You might give us more detail with your physical size/ frame size, etc. There is a lot of experience on C & V. If you got a problem, yo they'll solve it. (Check out the hook while my DJ revolves it.)

I apologize, I couldn't resist.

I find that I can ride in the drops on modern road bikes easier than I can our C&V bikes.

I'm not sure if it has more to do with geometry or the shallower drops that modern bars have. The older bars seem to have much deeper drops, hence it takes more reach.

Reach is quite variable in drop bars, especially vintage ones. Comparing a couple of popular examples, Cinelli mod 64 and the Soma Hwy One (compact) you lose only 3mm in reach.