There are two main differences between reading and computer glasses. The latter are designed to work at a longer distance. They also usually contain a material designed to reduce glare from the screen. One alternative to buying separate glasses for reading and computer use is to buy specially designed bifocals.
Note that there is a big difference between generic reading and computer glasses and those prescribed by an eye doctor or optician. Prescribed glasses will have lenses custom-built to fit the specific deficiencies of the user's eyes. Generic glasses often simply contain magnifying lenses similar to those found in a magnifying glass. While these may be suitable for many people, they will not be a suitable solution for everyone and could even do more harm than good.
Dedicated reading and computer glasses usually aim to serve different needs, most notably in terms of distance. This is easy to observe in practice. Most people read books with their arms bent slightly, meaning the book is around 18 inches from the eyes. For both visual and ergonomic reasons, the most common advice is that a computer screen should be at arm's length from the reader, which can be as much as twice as far away as book reading distance. This requires a dramatically different level of magnification and focus point.
Another difference with computer glasses is the need to counteract the effects of glare and reflection from a computer screen. This is a particular problem with older CRT screens, but is still an issue with modern flat-screen monitors. This can be tackled with either a light tint on the lenses or a special anti-glare coating.
Computer glasses lenses may also need to be specially designed to deal with the fact that the eye focuses in a different way with a screen. This is because, compared with the printed page, the contrast of characters to background on a screen is much less well defined. The lower definition causes the eye to need continuous refocusing, even when looking at the same character.
There are several solutions to these differences, besides simply buying separate reading and computer glasses. One is to have special bifocals, with one part of the lens dedicated to reading the printed page and the other part customized for computer screen use. This isn't an ideal situation for all users as, unlike reading a page from top to bottom, it's more likely the eye will need to dart about when using a screen, which can require head movement when using bifocals. A more sophisticated solution is progressive addition lenses that, rather than simply being split into two lens strengths, gradually increase power moving from the top of the lens to the bottom. This allows easier adjustment to different types of visual material.