Veterinary assisting is an entry-level position with no requirements in any state and are typically legally more limited in the tasks they can perform than veterinary technicians. This is how most people start out working in a clinic or on their way to becoming a veterinary technician or veterinarian. It's a great way to get in a clinic and start learning, not only about animals but also about the positions in a veterinary health care team so that you can figure out where you want to go from the assistant position. Because the majority of states now require that you be licensed/registered/certified to be a veterinary technician (and this is a growing trend), veterinary assistants do have limited potential for moving up in the clinic without earning a degree in something--veterinary technology, business management or veterinary medicine, etc.
Veterinary technicians have much more option for moving up and for finding positions related to veterinary medicine but with companies, research facilities, seats of higher learning, government jobs related to animal health and food safety etc rather than just in veterinary hospitals. Veterinary technicians can specialize in more than 10 areas after they are credentialed and that specialization makes them more sought after by specialty practices where their education and special skills are put to use daily. Veterinary technicians also typically make a good bit more than veterinary assistants. So, earning your degree in veterinary technology puts you in a better position for work in a veterinary facility but also gives you a much broader range of positions that will be open to you.