Whatsapp messages pass through their servers, but do not remain there. They are effectively sent from the sending device to the receiving device. This means that your messages reside on your device. When you use the desktop or web client, it's basically remote operating your phone.
This famously allowed Whatsapp to achieve massive scale with relatively few resources. [1]
Telegram messages reside on their servers. This allows for features such as those discussed above - multiple synchronized clients etc. The downside (aside from potential privacy concerns) is that it requires much larger amount of infrastructure, which presumably is one of the reasons they need $700 million.
[1] http://highscalability.com/blog/2014/3/31/how-whatsapp-grew-...