Veterinary technicians are required in most states to have a degree in veterinary technology from an American Veterinary Medical Assoc. Accredited collge and have taken and passed at least the Veterinary Technician National Exam if not a state board exam as well. They are also generally required to attend a set number of continuing education courses each year to keep up with changes in veterinary medicine. Veterinary technicians are educated in veterinary anatomy and physiology, pharmacology, animal husbandry, surgical assisting, anesthesia, medical nursing, diagnostics such as radiology and ultrasonography, clinical pathology, parasitology, medical terminology and record keeping, biological collection and sample handling and preperation, etc. They can also specialize in areas such as emergency and critical care, internal medicine, anesthesia, dentistry, behavior and equine nursing.
Veterinary assisting is an entry level position and there is no requirement for special education and training. Most training is done on the job and for this reason it is often very cursory and lacks the depth and breadth of a formal education. Veterinary assistants are generally taught the basic how-to but not the why or when you would do something different. They tend to do basic tasks such as animal restraint, basic care and sanitation, assist in patient monitoring, prepare instruments for use in surgeries or daily treatments, they may give medication as prescribed by the veterinarian, collect biological samples and perform basic diagnostic tests like reading fecals. Veterinary assistants are generally not the equivalent of a formally educated veterinary technician. For example, I know many assistants who can place an IV catheter and hook up a fluid line to it, but they generally don't know how to calculate the appropriate amount of fluids to give in a 24 hour period to maintain hydration, replace lost fluids from vomiting/diarrhea, calculate the appropriate number of drops per hour to provide the correct amount of fluids or understand the different types of IV fluids available and when each type is appropriate to a given situation. They may monitor anesthesia but they generally don't know how the different anesthetic drugs they are giving affect the body other than producing sedation or anesthesia---do they cause a drop in blood pressure that needs to be compensated for, do they make it more likely for animals that have seizures to have one, do they need to change the anesthetic protocol to compensate for heart, liver or kidney issues in a given patient. Veterinary assistants generally require much more supervision than a credentialed veterinary technician
However, there are some states which don't require any education to call yourself a veterinary technician and allow anyone off the street to perform any number of tasks on patients regardless. So pet owners need to find out who is caring for their pets. I have seen situations where owners thought that they were paying quite a bit for 24 hour nursing care of critically ill patients when the vet had hired an overnight person that they presented as a veterinary technician who didn't even know how to restrain an animal or assess their current condition and left that person alone in the hospital overnight. He was basically a warm body that could clean the clinic in the overnight hours.