I'm currently using a 4 character "strategy" myself. One has all the DoM classes, one has both melee classes, one is a Bard (and will take any future range classes), and the last has both tank classes. There are distinct advantages to using alts, but it does take some extra work in places.
Main Story This is a double-edged sword. The con is you'll have to redo the main story on each character. The pro is, you get to collect all the generous rewards from the main story on each character, something you couldn't do if you stuck to just one.
Achievements I'm not sure if this is a pro or con, but for unlocking achievements that require a great deal of effort, you may reach them faster (if it all) by sticking to one character. On the other hand, it might be more efficient points-wise to tag all the easy achievements on multiple characters. Since we don't know yet what the points can be used for, it's tough to say which is better.
Dailies / Weeklies A solid pro in favor of multiple characters is the extra helpings of dailies and weeklies you'll get, especially the Crystal Tower and collecting Myth Tomes. A single character will have a hard time gearing all 8 classes with CT and Myth Tome gear. For example, you're limited to 450 Myth per week on a single character and one drop from CT. But if you had 4 level 50 characters, you could collect up to 1800 Myth and 4 CT drops per week.
Cross-class Because of cross-class abilities, you do end up covering the same ground leveling up extra classes on each character, so that's one annoyance against multiple characters. However, if you know which class or classes your character will specialize in, it often isn't too much of a bother since most abilities are obtained at low levels and you can stick with leveling only what's necessary, and once finished, you never have to worry about it again (at least until they raise the level cap).
Suitable Racial Bonus Racial bonuses are minor, but if you level multiple characters, it's practical to tailor this bonus to that character's specialized classes. Every bit helps!