javax.annotation: @Nullable vs @CheckForNull

I think it is pretty clear from the link you added: if you use @CheckForNull and the code that uses the value does not check for null, FindBugs will show it as an error.

FindBugs will ignore @Nullable.

In practice this annotation is useful only for overriding an overarching NonNull annotation.

Use @CheckForNull in the cases when the value must always be checked. Use @Nullable where null might be OK.

EDIT: it seems that @CheckForNull is not well supported at the moment, so I suggest avoiding it and using @NonNull (also see Which @NotNull Java annotation should I use?).
Another idea would be to get in touch directly with the FindBugs developers, and ask their opinion about the inconsistency in the documentation.