what are the differences between Adobe Illustrator and Adobe Photoshop?

The main difference between this two programs is the way they create graphics, Illustrator creates and edits vector graphics and Photoshop creates and edit bitmap Graphics (photoshop can also handle vector graphics)

Vector graphics

Vector graphics consist of lines and curves that contain mathematical objects called vectors, Because Vector images are made up with mathematical formulas instead of pixels they can be scale up or down to any size without losing quality and in general they are much smaller in file size than vector graphics. It is important to note that You cannot use vector images for Web graphics unless the images are incorporated into special technologies such as Macromedia Flash.

Bitmap Graphics

As the name may imply, a bitmap image is made up of thousands of little color bits named pixels, Bitmapped graphics are measured by the number of dots per unit, usually called dpi (dots per inch). Bitmap graphics are much larger in file size than vector graphics which in turm will make your webpage slower to load and
occupy more hard disc space, also a unlike a vector image, a bitmap will change quality if you enlarge it or reduce it.

Having said all I will like to tell you that with both programs (specially Photoshop) you can edit create and manipulate both types of graphics, for example, Illustrator will allow you to convert a vector image to a bitmap and it will also allow you to import bitmapped images but you cannot edit them in the program.

If you are in the market to buy one of this programs I will recomend Photoshop, a more rubust program that will give you a lot more flexibility and options when it come to image editing.