Playing the F major chord on the guitar while singing the note E can sound wrong if you are not familiar with the sound of FM7 (F major 7th).
F major 7th chord = F A C E
F major chord = F A C
Add E using vocals and you have all the notes of F Major 7th and nothing extra. Though the vocal note is not there in the played chord, together, the triad chord and the vocal note forms a 4 note chord.
Playing the A minor chord while singing the note G can sound inappropriate if you are not used to the sound of A minor 7th.
A minor seventh has note of A minor + the G note
Why would anyone play the F minor chord while singing the D note which is not in the chord? Together they give you the F#minor 6th chord. Since it has very similar notes to the Bb7th chord, the chord could act as fifth chord to go back to the root D# chord. Instead of playing the 7th chord, we use a combination of a minor chord and a note on the vocals to play a 4 note chord similar to the 7th chord.
Mark Levine states in his The Jazz Theory Book, how there are no wrong chords. On a more basic level, it about 'sorting for relevance' instead of 'sorting for good or bad, right or wrong'