I haven't stayed in any hotel in Beirut as we have an apartment.
However, I visited these two hotels many times and know people who've stayed in them.
The Le Vendome is a nice charming hotel. It's small and more personal perhaps. It's a boutique hotel.
The Phoenicia on the otherhand is a large business-oriented hotel.
It all depends on what you are looking for.
The two hotels are very close to each other. It's just a 5-10 minute walk between the two.
I'm not too surprised that you can't find any availability in the Le Vendome. It gets filled up very quickly as it is quite popular and at the same time very small. It only has 73 rooms versus the 446 rooms in the Phoenicia.
Regards