A wizard in the HP universe, seems to be someone who can harness magic in some way or the other. Using them with a wand is only one technique of harnessing it. Potions, divination etc are other ways of doing so. What one could consider is that, spells made using the wand are a direct means of doing so.

The most talented wizards (Harry, Voltemort) and others through practice can channel magic directly in a raw unfocused manner. As someone said here, it is an emotional release of inner potential more than anything else.

They aren't spells though, but for people who understand how magic works and the science behind it completely or magical geniuses / savants (harry potter), can use raw magic to perform as an application as to perform the same end results that an equivalent spell could provide.

Wands are common practice for half-bloods / common wizards do the same what their savant counterparts can. Since not everyone can process and apply the whole manual process, they have focusing tools like the wand and spells (also because of the vast variety of things they allow).

Think of it like this, an olympic athlete can run a five mile course in a very rough terrain in the same time it would take someone to take a car in the same. Yes, the athlete could take a vehicle, when needed they do, but in Olympic games, where you cannot drive cars on tracks, they run.