There's only a gap of 21 years between the two films, but in the interim there's been a revolution in Shakespeare production.
It's all about text. Olivier reverences the poetry. He and his cast take it so slow they make it sound fustian. The slow pace necessitates swingeing cuts. Richardson respects Shakespeare the playwright- and crams in as much of the text as he can. His cast go at their lines so fast you forget you're listening to Elizabethan English. They're naturalistic. They gabble.
(But surely Shakespeare's actors must have taken it fast too. If you performed the full text at Olivier's pace it would last four to five hours.)
Olivier is cinematic. Richardson adopts a televisual style, shooting in close-up against bare brick .
Olivier is expressionist, Richardson is kitchen sink. Olivier's Hamlet is a renaissance warrior, Nicol Williamson's is a scruffy, squeaky-voiced student. Olivier relishes the supernatural, giving us a terrific ghost and lashings of popular Catholicism, Richardson hates all that- his ghost is a white light, he cuts Hamlet's soliliquy about killing Claudius at his prayers.
Olivier- by cutting and rearranging the text and through imaginative directorial touches- imposes meaning on the play (it's all about sex). Richardson- using a fuller text and letting the actors do the work- respects its ambiguity.