There is no difference. A twister is a tornado. The two words describe the same phenomenon, and no meteorological body draws a distinction between them, defines a threshold that separates them, or uses one to mean a weaker version of the other.
So why do two words exist?
Because one is technical and one is vernacular. Tornado is the term used in every official context: the National Weather Service issues tornado warnings, the Storm Prediction Center keeps a tornado database, and the rating scale is applied to tornadoes. Twister is informal American English, older in everyday speech than the official vocabulary is in most people's ears, and it survives because it is vivid and short.
The 1996 film Twister then fixed the word in place for a generation, which is why the question gets asked at all. A great many people first encountered a tornado on screen, under a title that used the other word, and reasonably concluded the two must denote something different.
Does the National Weather Service ever say twister?
Not in a product. You will not see a warning, a watch, a storm report or a damage survey use the word. If you are reading something that calls it a twister, you are reading journalism or conversation rather than an official record. That is a useful tell when you are trying to work out how authoritative a source is.
What terms do describe different things?
Plenty, and this is where the real distinctions live. These are not synonyms for tornado and getting them wrong changes the meaning:
Funnel cloud. A rotating column of condensation that has not reached the ground. It becomes a tornado at the moment it makes contact and produces damage, and not before. We have a separate piece on telling a funnel cloud from ragged scud, because most of what gets photographed and reported is the latter.
Waterspout. A tornado over water. Fair weather waterspouts form by a different mechanism from supercell tornadoes and are generally much weaker, though a tornadic waterspout is simply a tornado that has moved offshore.
Landspout. A tornado not associated with a rotating supercell updraft. It forms from the ground upward rather than descending from a mesocyclone, and is usually weak and short-lived.
Gustnado. A brief spin along a thunderstorm's outflow boundary. It is not connected to the cloud base at all and is not a tornado, despite looking like one from a distance.
Dust devil. Not a storm phenomenon in the same sense. It forms on a hot, clear day from surface heating with no parent thunderstorm involved.
Is a twister weaker than a tornado?
No, and this is the specific misconception worth killing. Because twister sounds casual, it is often assumed to describe something smaller. It does not describe anything. The 1925 Tri-State tornado, which killed 695 people over a track more than two hundred miles long, would be called a twister in ordinary conversation and a tornado in the record. Same storm.