First of all, yandex is a very old search engine (launched in 1997). For a very long time yandex was way better for search queries in Russian which was obviously a deal breaker for domestic users. Like google and yahoo, they have a full suite of internet services (mail, search, news, virtual currency, etc), so you wouldn't gain much by switching.

Yandex also possesses some unique domestic services that google doesn't have (market, taxi, traffic jams).