The vSphere Web Client is quite “infamous” for its dependence on Flash and its performance related issues. However, the fact remains that the Web Client is the only way to manage the new features on vSphere. Now having said that, it doesn’t mean the old vSphere client is dead.
So the question arises: when to use which client. Below table should make it more clear.

vSphere Web Client only

SSO

Tags

Ability to pause tasks

Saved searches

vSphere Replication

Extended Menus for VCO

VDS Health Check

VDS Export/Restore Configuration

Log Browser

vSphere Data Protection

vSphere C# Client(Desktop)

Fully functional VUM plugin

Various 3rd Party Desktop plugins

VXLAN Networking

vCenter Server Maps

Connect to an ESX Host

Use the datastore browser to inflate a thin disk

VMware is planning to move to a HTML5 based web client but it doesn’t look like its going to be released anytime soon. So till then, we will have to live with the web client :)