Read the article of the Support Centre
In the following, I'll be only explaining things you can not find within this article.

Fmods can close threads. This is either because the rules have been broken or because the thread creator wants his thread to be closed.Fmods can make normal threads stickies.
Toxic posts can be hidden for normal users by Fmods. For their colleagues and Jmods these posts are shown red (and its content is shown.* Threads that are in the wrong forum can be moved by Fmods. In the original forum there then still is a blue hint with a link to the thread in the new forum.
Fmods can change thread titles. Concerning bumping, Fmods can see how often a thread was bumped (and Fmods can bump every thread not just their own. Therefore there is no 10min rule for Fmods. )*
For bigger or many rule breakings Fmods can ban a player. (The durations they can choose are 24, 48 and 72h. Regularly you get 48 hours. )* The ban then will be redirected to a Jmod who then decides what action is taken. For example, the Jmod could additionally apply an in-game mute or of course, unban the player.
For discussions, they also have access to Fmod forums (and a forum that both, Pmods and Fmods, can use)*. To discuss important things Fmods can easier contact Jagex Mods.
Threads that need Jmod attention can be reported by Fmods (together with comments and/or questions. After reporting a thread Fmods can receive feedback through a system.*
Fmods have a say when it comes to new Fmods. However, if a person is picked or not is, in the end, the decision of the responsible Jmod. During meetings with Jmods Fmods can, for example, discuss new forums before they are created. They can not change anything directly but they can bring up pros and cons for or against the forum.)*