Who have Vanquish? I need help!

Steam has a habit of swallowing error messages if a game cannot be started for some reason. Go to the game's install folder and start it with the .exe directly, maybe that will give you an error message that could help.

I had no issues running it myself so I can't really help other than suggesting you try to verify the game files.

I try but nothing :/

Did you tried restarting the PC?

When a Steam game doesn't start, these are some things to try:

1. Troubleshoot -- If the hang up is a Windows issue, this will catch it. (video)
2. Reinstall -- Steam doesn't always download everything correctly, so try again. (video)
3. Update Drivers -- If the graphics or sound drivers are bad, this will fix them. Remember to restart your computer.
4. Update Steam -- Occasionally, the problem is Steam, not the game.
5. Verify game chache -- If there is a corrupted file, this will fix it. (video)
6. Run as Admin -- This is rarely the culprit, but it doesn't hurt to try.

That's a relatively common issue with steam. Have you've tried restarting steam, and if that did not work, restart your computer?

I wonder if it installed all the prerequisites along with the game...
Go to your game's folder, then into the "redist" folder & install each of the items in there (DX,/ VC / PhysX / whatever)
Reboot & retry

One possible problem in such mysterious situations...Firewall or AntiVirus program.
Sometimes these are so kind and don't notify you about something they blocked :)
I almost doubt this to be the reason for your problem, but it never hurts to check these two.

Steam

"I have Sonic part of the ROG Suite installed when I got my MB. It was running in the background so I switched it off and the game runs...amazing and weird"

\o