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 consciously used for ads.
Keywords therefore only exist in manual campaigns, where each keyword can have their own bid and match type (broad, phrase or exact).
For broad and phrase ads, sponsored products will not, however, always be found via the actual keyword that was targeted, but via other searchterms.
Note: Since October 2018 ASINs can also be targeted.
Searchterms are those word-combinations which customers actually put into the search bar.