Here’s the official difference from Merriam Webster’s Dictionary of Synonyms (a great book for picking apart nuances):
"Exculpate, absolve, exonerate, acquit, vindicate mean to free from a charge or burden.
Exculpate implies simply a clearing from blame, often in a matter of small importance <directly Harding was blameless for what was going on. Indirectly he cannot be wholly exculpated–S. H. Adams>
…
Exonerate implies relief, often in a moral sense, from what is regarded as a load or burden … In general exonerate more frequently suggests such relief from a definite charge that not even the suspicion of wrongdoing remains <exonerate a person charged with theft>."