Again… both programs share many of the capabilities of the other. Photoshop can create “vector objects” like shapes and typography. However they become part of the “bitmap” once rasterized. (Once the file is “finished”.) Illustrator offers painting, brushes, colorizing and can even import and manipulate “bitmapped” images — however these are all “vector objects” NOT bitmaps. So it can become confusing.
Most professionals and higher-end hobbyist prefer to own both programs because of their unique capabilities which complement each other.
If you need to acquire, modify, print or deploy to the web images of a “photographic” nature, then PS or PSE would be the program of choice. If you need to create complicated graphics like maps, logos, schematics, or even publications for print like ads, brochures, menus, etc, — particularly with typography — then Illustrator is the one.
The above is a very basic explanation. Both programs have so many features one would be at a loss to decide which is “best” … they’re both “best”. The projects and activities one encounters most often should dictate which software would be the most beneficial if only purchasing one. Saving Money?
For what you described above, Photoshop Elements is probably the most ideal for your constituency. It’s available for under $100 from many vendors, and even less as a bundled product with scanners, digital cameras, etc.
You did not say which operating system you’re using, but I’m assuming it’s Windows since the largest percentage of emergencies we field are from Windows users. If you’re using Macs, download a copy of GraphicsConverter. It’s a shareware Photoshop “surrogate” of sorts. There are many others for Macs and Windows — however there are NO surrogates for Illustrator.