The difference between a veterinary techician and a veterinary assistant are the education and credentialling requirements.
No state in the US has any educational requirements for working as a veterinary assistant. This is an entry level-position in a veterinary facility and training is generally done on the job. This training 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, many assistants 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 etc. Or when a typical anesthetic protocol may cause increased risk due to medical conditions or breed specifics.
There are voluntary educational opportunities, however these are not equivalent to a college degree program and are instead basic vocational training. There is no over-sight by a professional body to ensure that the majority of these programs provide adequate or correct information. There is no requirement for hands-on training and instructors often have little or no experience or education in the veterinary field. There are a handful of certification programs that are designed and approved by veterinary professional organizations or that are offered by colleges which also offer accredited veterinary technology programs and these are better choices for someone who wants to be a veterinary assistant. All of these programs offer certification as a veterinary assistant, but this certification is not legally recognized or required by any state in the US.
Veterinary technicians are required (in most states) to have a 2 year degree in veterinary technology from an AVMA accredited veterinary technology program, to have passed the Veterinary Technician National Exam and a state exam in order to be credentialed. 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. The American Veterinary Medical Association maintains a list of accredited degree programs on their website: http://www.avma.org/education/cvea/vettech_programs/vettech_programs.asp
In some states, the use of the title "veterinary technician" and the practice of veterinary technology is recognized as profession and licensure is required. In other states, veterinary technicians are registered or certified. The laws that govern veterinary technicians vary from state to state so for specific information on the laws a person should check their state veterinary practice act or contact their state veterinary licensing board.
A very general list of things that a veterinary technician would do would include collecting patient histories, collect biological samples (blood, urine, feces, etc), running diagnostic tests, monitoring and medicating hospitalized animals, assisting in surgery, administering and monitoring anesthesia, performing dental cleanings, providing treatment for outpatients as prescribed by the attending veterinarian, filling prescriptions, answering client questions on preventative medicine, disease processes, medications, etc, maintaining inventory, caring for surgical and medical equipment such as anesthesia machines, taking radiographs, entering medical records, etc.
Veterinary assistants tend to make close to minimum wage. Veterinary technicians will generally make more due to their education and increased abilities.