Well first things first, to understand the differences between gaming consoles and computers you also have to know whether the computer is used primarily for gaming or not. If not, then the obvious difference is that one is meant for computing with some gaming being a benefit, whereas the other is mainly meant for gaming.
Now let's talk about some of the simularties of a computer and gaming console. Both xbox 360 and the PS3 are essentially computers, in fact some of the same components in the 360 are used in PCs. The PS3 is a little different in the sense you won't find most of the hardware in the PS3 in any PC. Accessories for each system differ in their connection mainly. The 360 for instance uses proprietary wireless technology for it's wireless connectivity of it's wireless controllers, and accessories. The PS3 however, uses industry standard Bluetooth for it's connectivity to wireless accessories.
Although you stated Playstation and Xbox, I am assuming you're referring to the current generation of gaming systems which are actually Playstation 3 and Xbox 360. (There was a previous version of the Playstaion called 'Playstation' and the original xbox was called 'Xbox'.)
Can a gaming console be used in place of a computer for other normal computing jobs? ABOSLUTELY, the PS3 can be loaded with the Linux operating system, and you can then install Firefox (web browser), Open Office (Office apps, similar to MS Office), and other Linux based applications and then allow the PS3 to boot into Linux with the provided boot loader. In fact Yellow Dog Linux was the first to get a distribution of Linux to work with the PS3, there are now others.
Here's the thing, unlike Microsoft and the Xbox 360, Sony shows you how to replace the hard drive with a bigger drive, and how to load the OS on your PS3, without voiding your warranty. That's right, putting Linux on your PS3 DOES NOT void your warranty, nor does putting in a bigger hard drive.
The PS3 has usb ports, so that you can use usb devices, such as keyboards, mice, webcams, etc....
Although the 360 also has usb ports, keyboards and mice may work, but webcams, not likely to work. Also you cannot install an additional OS on the 360.
Hope this helps a bit.