Why is Lebanon is drawing 10 million cubic meters/year from the little Wazzani rather than the mighty Litani? Or rather ... what's the reason given by the Lebanese? According to PM Rafic Hariri's official website, the UN and US suggested using the Litani ....:

But the Prime Minister added, from "day one," he made it clear that the Litani River hardly satisfies the demands of the region and the Wazzani waters are needed for vital projects. Mr. Hariri said while Lebanon entrusted the United Nations with the issue, the country has also been in contact with the United States, the European Union, and the international community, to counter Israeli claims.

Haririi says the Litani "hardly satisfies the demands of the region", but the Litani discharges about 700-900 million cubic meters per year into the Mediterranean Sea (source).