For example, if I’m playing a song in G major on the piano, accidentally playing F natural immediately sounds off, but when the composer indicates to play natural, it still flows. What’s the difference?
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Some intervals are more dissident than others. The order from least to most dissident is as such
Octave X
Major 5th. X
Major 4th. X
Major 3rd. X
Minor 6th
Minor 3rd
Major 6th. X
Major 2nd. X
Minor 7th
Minor 2nd
Major 7th. X
Tritone
The one with X is in the western major scale. Notice how they’re on the less dissident side. Including a minor 3rd or Minor 6th in a melody wouldn’t sound too out. Adding the Tritone would feel quite jarring.
Edit, with your example, an F is the Minor 7th in G. These can sound good if you make a Gm9 chord with G Bb D F A for example. Also a dominant G7 chord would sound good whilst resolving to C
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