An assault rifle has a well-known definition. They are select fire, use intermediate cartridge, and feeds from a detachable magazine. They are highly restricted and unavailable to civilians unless they were registered as an NFA weapon before 1986.They are very expensive for non-government purchasers.
An assault weapon has many, poorly thought out definitions. Generally speaking it's a semi-automatic rifle with any number of mostly irrelevant characteristics (feeds from detachable magazine, threaded barrel, bayonet lug). These are legal to own in most states and smaller jurisdictions, but obviously one's situations may vary.