I don't use AI or PS, and though I've heard PS can do some vectoring if it involves fonts, if your logo is more than pure text (any image parts) then I'd prolly open illustrator, import the plain old raster from PS, and see if you are better off either doing a bitmap trace or building the logo itself in Illustrator (depends on what the logo's like).
I use Gimp and Inkscape. I had made a bunch of logos in GIMP using Impact font of two colours, so, pretty much text, and then there was a little bit of image (a wheel replacing the letter "O"). What was easier than playing with the GIMP-made PNG was just to open Inkscape, type my text out in Impact font and change colours etc, then I imported the wheel.png and did trace-bitmap to get a vector version of the wheel. I deleted the bitmap wheel and moved the vector wheel where I needed it. In other words, I re-made it from scratch in my vector program because for the type of logo it was, this was just easier.
If you have a SRLSY complicated logo though then it can't hurt to see what happens if you save as .eps in Photoshop and then, to see what you really have, open that file in Illustrator.
An .eps is basically like a PDF. PostScript language is used to make the image. But you may for example end up with a larger than necessary file importing a raster into Ai or trying to just save as .eps (I'm not sure).
Whenever possible I try to make any vectors as vectors in the first place, because then I'm more sure I have the cleanest, leanest possible file (barring any goofs I make in the program lawlz).