In harmonic intonation, the intervals tend to compact, while in pure unaccompanied melody, the intervals tend to stretch. (This statement oversimplifies things, but it captures the essence of what happens.) This means singing involves a constant balancing act between harmonic and melodic intonation.
For example, about fifteen years ago, my wife and I went to the annual First Night Celebration at Rockport, MA. The town hosted a number of events: magic acts, kid’s shows, music, etc. One event we went to featured a barbershop quartet.
For those of you not familiar with them, barbershop groups contain four men who sing without instrumental accompaniment. This allows them to tune their chords better than they would if they had a piano playing with them. (The term Barbershop probably comes from the time when barbershops acted as community centers, including hosting board games and amateur singing.)
I vividly recall that at one point in the barbershop performance they had a tuning mismatch between the melody and harmony. It happened when they switched from one chord to the next. As they brought the new harmony in tune, they pulled the top melodic interval out of tune.
At the time, it felt to me like they had stretched the melody like taffy. This is when I realized you can be harmonically in tune and melodically out of tune. (The guys in the video above, though, do it right.)
Conversely, when singing with the piano, the melody will sound in tune, but the harmonies will sound at the edge of being in tune. This gives an alternate example of how harmony and melody tune differently that many singers have experienced for themselves.