If when you click the Play button in the launcher and no game window appears (but Warframe.x64.exe appears in task manager), please try the following:
First ensure any lingering processes are terminated by entering the Task Manager (Ctrl + Shift + Esc) and under the Processes tab, highlight the Warframe.exe entry and click End Process.
Next, attempt to launch the game.
This document will be updated if new information on this subject arises.