Can you keep a free weekend game?
The temporary license ends with the promotion. Keeping the game installed does not give you a permanent license. You need the full game’s license to continue playing. Steam’s official Free Weekend documentation explains how this access works.
Is it the same as free to keep?
No. A free-to-keep promotion gives you a permanent license for the offered product if you claim it during the promotion. A free weekend only provides temporary play access.
What about demos?
A demo is a separate trial version. A free weekend instead temporarily opens the base game. Check the product and the exact event wording on Steam before downloading.