In most cases these terms can be used as synonyms.
In the field of Sponsored Products (PPC) they are, however, defined as follows:
Keywords are word-combinations which were conciously used for ads.
Keywords therefore only exist in manual campaigns. Here ads can be placed with different click-price bids for each keyword.
For broad and phrase ads the sponsored product will not, however, always be found via the actual keyword, for which the ad was placed on, but via other searchterms.
Searchterms are those word-combinations which customers actually put into the search bar.