What is the difference if a note is marked with forte (f) or if it's marked with marcato (^)? Would it sound differently?
The forte symbol: f means "loud", but it doesn't mean just play that note loudly; successive notes after this symbol should also be played loudly, (until another dynamic symbol is encountered).
The marcato symbol: ^ means "marked" or accented. This symbol only effects the notes it is on, and successive notes should not be accented.
The sforzando symbol: sfz is another dynamic mark that means "forced". This symbol only effects the note it is placed on. Sforzando is very similar to a marcato.
Marcato can be played with a dynamic of a p (piano). As you say it is an accent relative to other notes in the same, previous or following bars.
Forte is depending of the velocity of the attack so it implies for keyboards somehow an marcato, but with other instruments (strings, woodwind and brass) you can play forte with more or less marcato.
Marcato is an emphasis. Particularly on longer notes, it's very much like a sfz in that the note reverts to the current dynamic level after the initial emphasis.
Forte, in the absence of crescendo/decrescendo marks, indicates a constant sound level for the entire note (as well as all notes following until other markings appear).