These two classifications seem to point to the same types of words.
They are two different distinctions that, however, often go together.
"Open vs. closed" is a diachronic distinction meaning how easy a set of expressions can be extended by new elements.
"Functional/lexical" is a synchronic distinction referring to the role played by the expression (e.g. serving as a full verb or as an auxiliary).
Furthermore, the functional vs. lexical distinction is a theoretical concept and depends on one's theoretical assumption, whereas "open/closed" is more of an empirical observation.
As jlawler said, the two distinctions often amount to the same expressions as being a functional class makes it less likely for other elements to enter that class because of the important role they play for the grammar and their grammaticalized status.
However, there are instances, in which they two distinction do not line up. For instance, prepositions are considered to be lexical categories by many, but they are a rather closed class (for a discussion of the various positions see this article by Joost Zwarts.)
NB: Even if it is true that "functional expressions" have a more abstract, i.e. more grammaticalized meaning, they often still have meaning; e.g. the tense/mood expressed by the auxiliaries in English or many other languages. But there are also cases in which the meaning is more or less lost [do-support with negation, for instance]