This came up in the comments thread of this question so it was suggested to ask it.
Spent means the sentence etc has been served and, under certain conditions, one need not disclose one's criminal history as they are deemed rehabilitated. See the Rehabilitation of Offenders Act 1974 and NARCO:
Once a conviction, caution, reprimand or final warning becomes spent, you do not need to disclose it to most employers, or when applying for most courses, insurance or other purposes (e.g. applying for housing)...
Expunged means the conviction etc has been quashed as though it never existed. The online OED defines it as:
Obliterate or remove completely (something unwanted or unpleasant)
Note that the term used by in legislation is Quash[ed], as per section 2(2) of the Criminal Appeal Act 1968:
In the case of an appeal against conviction the Court shall, if they allow the appeal, quash the conviction.
A spent conviction has had its punishment served.
An expunged conviction is one that has been served and is so old (I.e. there is a statutory time limit) that it can no longer be reported for particular purposes. Usually on a police record although there may be different grades of police record - you maybe expunged in some but not on others. For example, certain convictions may not appear on publicly available records but may be available to police or courts.