The concise Oxford English dictionary defines "verify" as "to make sure or to demonstrate that something is true, accurate or justified", from the medieval Latin verificare, from verus ("true"). The meaning of "valid" is given as "actually supporting the intended point or claim" from which "validate" is "to confirm the truth or value of". The Latin origin is validus ("strong"). Verification is about truth and accuracy, while validation is more about supporting the strength of a point of view or the correctness of a claim.
The use of the term verification in meteorology has a long history. Finlay used the term "verification of predictions" in his 1884 paper, and it was then used by later authors. Outside the meteorological community, the terms "evaluation" or "assessment" are more frequently used than "verification". The meaning in engineering is described in a Wikipedia article.
In the context of the above definitions, the meteorological use of the term verification, which is about "accuracy" or "truth" of the forecast compared to observations, would seem to be consistent with the dictionary definition. Verification connotes objective comparison of forecasts and observations to establish accuracy and truth of the forecast. Validation tends to focus more on the process used to make the forecast. It might be a more appropriate term to use for example when checking that a model change produces results that support expectation. That is, does the new version of the model work as it is supposed to?
One might even speak of "validating the verification system" which would mean making sure that the methodology used to verify forecasts works correctly, producing the intended results. Validation is a check on the correctness of a methodology; verification determines whether it produces accurate results. Validation is a more general term, less quantitative than verification.
Put more briefly:
to validate is to check that one is doing the right things,
to verify is to check that one is doing things right.