I believe Riker (and Troi, for whatever reason) often oversaw the actual crew as seen by the crew evaluations and such that are seen throughout the series, which Picard isn't involved in. I would imagine this is relatively consistent with how it would operate in the real world as well.
Glad my meaningless question thread managed to generate some meaningful discussion. I may post a separate thread about this because I now have several questions about who selected which crew members for assignment and promotion.
My failing memory actually did recall that Riker oversaw crew evaluations. My assumption was that he is responsible for transfers and promotions, but it's got me thinking: who selected the original, starting crew for the Enterprise? For example, did Picard select the senior staff and only other key positions?
Follow-up question: if Riker was responsible for most transfers, assignments, and promotions, I take it Picard still had to approve each one of these?