As a commenter suggested, when we use fabricate in the context of deception,1 we imply that some effort went into inventing or producing something disingenuous, either a story or an artifact, like a fake document. If you simply said 'no' when you knew full well that the truth demanded 'yes', it is unlikely anyone would say that you 'fabricated' an answer. But everyone would agree that you lied.
1In other contexts, fabricate can mean simply to invent, create, or to construct, manufacture; specifically, to construct from diverse and usually standardized parts (Merriam-Webster).
On a formal grammatical side, fabricate is a transitive verb, while lie is most commonly intransitive.2 Normally, we simply say that e.g. he lied, or else we add a preposition phrase and say that he lied about [something]. In contrast, we say that he fabricated [something], as in he fabricated a story.
2Much more rarely, lie can also function as a transitive verb. Here is an example from Merriam-Webster: He lied his way out of trouble.
As far as the meaning, fabricate means 'To concoct in order to deceive' (fabricated a convincing excuse) (American Heritage); 'To "make up"; to frame or invent (a legend, lie, etc.); to forge (a document)' (Numerous lies, fabricated by the priests..were already in circulation; If any person..wilfully fabricate in whole or in part,..any voting paper.) (OED).
As for lie and similar words, Merriam-Webster has the following useful discussion:
lie, prevaricate, equivocate, palter, fib mean to tell an untruth. lie is the blunt term, imputing dishonesty. // lied
about where he had been // prevaricate softens the bluntness of
lie by implying quibbling or confusing the issue. // during the hearings the witness did his best to prevaricate // equivocate
implies using words having more than one sense so as to seem to say
one thing but intend another. // equivocated endlessly in an attempt
to mislead her inquisitors // palter implies making unreliable
statements of fact or intention or insincere promises. // a swindler
paltering with his investors // fib applies to a telling of a
trivial untruth. // fibbed about the price of the new suit //