This is a great question. You can think of a veterinary technician as a veterinary nurse, while a veterinarian is the doctor. In the clinical setting, a veterinarian is responsible for establishing a relationship with the animals owner, examining the animal, deciding on the appropriate diagnostic tests to best investigate the animal's health, interpreting the results of those test, prescribing the treatment and keeping records of the animal/patients health, test results and treatments. Some veterinarians perform surgery. A technician works along side the veterinarian to help the vet carry out the responsibilities mentioned about. Depending on the type of practice, the technician may be the one who runs/performs the diagnostic tests --from collecting samples of blood and feces to preparing these sample to be sent to the lab or even analyzing the samples in the clinic-- taking radiographs (x-rays), administering medications, giving vaccination and other injections, and assisting the veterinary in procedures like dentistry and surgery. Both jobs work to promote the health and well being of animals. But the vet and vet tech need to know basic biology, chemistry and mathematics; vets go through more extensive schooling to learn about practicing medicine, treatments and surgery. Both vets and vet techs, though the work a lot with animal, also need good communication skills and interpersonal skills as there is always, in some way, shape or form, a human attached to the animal being treated.
Remember this a very broad overview, because veterinarians and vet technicians work in a variety of fields and environment -- dog and cat clinics, farm vets, research, zoo and wildlife -- the specific task and responsibility can be just as varied.