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They are both means to accomplishing an end (communicating with a database). ADO can use both. OLEDB is the, current, access method of choice. ODBC is older technology that will, eventually be phased out.
At this point in time, I would use ADO and select the access method that works best for you (but use ODBC only if you cannot, or with great difficulty, accomplish the task with OLEDB).
Well, in my DB book it states that OLE DB is the next generation of Data Provider after the older ODBC. ODBC was designed to specifically work with SQL as the query language; however, OLE DB is supposed to be able to accept any query language supported by the Data Provider. Thus, according to this, you should be able to use OLE DB to connect to all sorts of weird things like mail folders and system directories, etc.
It looks like the bottom line is that OLE DB is supposed to be better and more efficient (since its newer), so use it first; use ODBC only if an OLE DB Provider is not available for the Data Source you're trying to connect to.
Hope that helps!
The thing you have to remember is that, relatively speaking, OLEDB is brand new. That generally means there are a few kinks to work out. ODBC, on the other hand, has been around for years and most of the significant problems have already been worked out.