The examples of Harry doing the wandless magic with the snake and blowing up his aunt are actually good ways to explain why wizards, specifically young ones at Hogwarts, use their wands. The wandless magic Harry uses is sloppy, unpredictable, and mostly unintentional. Harry can't do wandless magic "on command". Only powerful and fully trained wizards, Dumbledore for example, can do wandless magic. The wizards use the wands and the spell words to direct their magical energy into an intentional and predictable action. If you'd like, the wands are like training wheels. Plus, wandless magic probably requires a lot of energy and tires the wizard faster than if he/she had help channeling their magical energy.
(Related to the question: Harry only does wandless magic when he is angry with someone, much like how Tom Riddle made "bad things happen" to people who were mean to him. This is an ability which, like parseltongue, was 'given' to Harry with Voldemort's soul, so he wouldn't necessarily be able to perform it if he were not the Chosen One.)
For more information, check out the Harry Potter Wiki page on wandless spells, which backs up the points I made.