In most cases, these terms can be used as synonyms.
However, in the field of Sponsored Products (PPC) they are defined as follows:
Keywords are word-combinations which were consciously used for ads, originating from you (the Seller/Vendor).
Keywords therefore only exist in manual campaigns and are what you bid on so that your products get found. In other words, ads can be placed with different click-price bids for each keyword.
For broad and phrase ads, the sponsored products will not, however, always be found via the actual keyword, for which the ad was placed on, but via other searchterms.
Searchterms are word-combinations which customers actually put into the search bar, originating from the customers.