Why is my character out of date?

The quick version:

Logging into and then logging out of your WoW character should cause the Blizzard API to update.
It usually takes a few minutes for the Blizzard API to update, sometimes it can take longer.
In some cases, the Blizzard API is just busted and will not update your character. This is on Blizzard's side and any 3rd party tools that use the API will not be able to see the current status of your character.
When possible, use the SimulationCraft addon. It's better than "Load from Armory", doesn't require logging in / logging out, and is far more reliable.

The detailed version:

The Blizzard API is provided by Blizzard and is used by the Raidbots Discord bot and the "Load from Armory" feature on the website. The API is a set of tools that 3rd parties can use to get information about the game, characters, guilds, and more.

Information about your character is supposed to be updated whenever you log that character out of the game. The spec, gear, talents, and other information is then available to be used for sims.

Unfortunately, the Blizzard API is far from perfect. From time to time, the API will not be updated when you log out and so 3rd party sites will not be able see the current status of your character. These are problems on Blizzard's side and unfortunately there's nothing that I or any other 3rd party sites can do about it. You can try to logout again to see if that updates your character.

The only workaround is to use the SimulationCraft addon on the Raidbots website. This is a better option anyway - you do not need to logout when changing your character and it cannot be out-of-date like the Blizzard API.

Raidbots cannot find my character at all.

Check your Blizzard Account Privacy settings and make sure that "Share my game data with community developers" is Enabled.

The Armory website shows my character correctly but Raidbots has old info. What's up?

Blizzard's Armory website is different from the API they provide to 3rd parties. It is somewhat common that the Armory website is up-to-date but the API for 3rd parties is not up-to-date.