Photoshop, PaintShopPro, Gimp, and Fireworks are all bitmap graphics programs. This means their output is images mapped out in a pixel-by pixel grid, which is directly relevant to images used on the web. Although some of them have vectoring abilities, when rendered as an image the vectors are drawn into a bitmap layer.
Learning a vector program such as Illustrator or Freehand is an invaluable tool when designing graphics (Freehand is probably not supported any more, since it's been bought by Adobe.)
Since you said "logo," let's take a scenario: You design a logo for a customer and put it on their site. They love it so much they want to use it on their business cards, maybe even a full color brochure.
But the web resolution of 72 DPI is too low for any of these applications. Sizing it up will still look pixelated and poor. You basically have to recreate the logo at a higher resolution.
Inversely, thinking ahead, your original may already be in a higher resolution, but when reduced you lose a lot of detail in type and fine lines.
If the logo is designed in a vectored program, you just export it at the resolution you need. Vectored art is resolution independent and you just draw it on the device at the resolution you need (The "device" can be a file, as in, export as a 72 DPI jpg.) So instead of spending 3 hours on getting a quality billboard sign out of a web page logo, you spend five minutes tops.
I just had this last week with a client, she needed a version of the logo in CMYK for printing, spot colors for a 4' X 4' sign on her trailer, and another for 4" X 2" logos on caps - all from the same file without reworking any of them.
Vectored programs are invaluable if you work with Flash as well. So don't ignore these points, a little experience working with a vectored program is an invaluable resource in any designer's toolbox.