[tbw2]

It would make Russia look good to the remaining secular dictators by propping up Assad over the Jihadis.
It disrupts expansion of Jihadis who have been causing problems in Chechnya for years.
Protecting the Tartarus base keeps them relevant in the Mediterranean.
It keeps Russia relevant in world affairs. They could even say they are the new power in the Middle East since the U.S. is withdrawing from Iraq and Afghanistan and made things worse in Libya and Egypt by tossing missiles and letting extremists take over.

[ETL]

They could even say they are the new power in the Middle East since the U.S. is withdrawing from Iraq and Afghanistan

Bingo! That has been Obama and Putin's plan all along. Part of their "new world order".

[Vendome]

In the new world of geo politics everyone is just happy to have the Good Ole USofA as the sole superpower. This leaves them free to exert their influence in the sphere they choose. Think China and the supposed South China Sea, India and it's "territories", Russia and Iran as a quasi confederation. France and Germany doing whatever they can to remain relevant on the world stage and using American power and influence to force change in Libya, Syria, South Africa, etc. All areas we (read I) could care less about. This also mitigates against their spending cash on any real offensive forces, as a percentage of their GDP, as a opposed to our ability and desire to spend upwards of 4% of GDP for the latest in military gear and offensive forces.