An interval is a measurement; a way to analyze music theoretically. A chord is an actual musical component. If you "play intervals", you could technically be said to be playing arpeggios, which are "broken chords".
Many definitions of a chord require a third note, drawing from Euclidean geometric definitions: "two points define a line; three define a plane, four define space". The parallel definitions would state that two notes define movement of pitch, but three are needed for a chord in order to define its quality (major, minor, suspended, augmented, diminished). However, two notes, especially two lines of notes, can indicate the quality of a key in which the performers are singing or playing.
For example, listen to this:
There Is No Rose - Traditional Carol
The full choir is in more or less three-part harmony; though there are four parts, two or more double on each sustained chord. But then the middle portion drops to a duet. With the tonal quality of the piece defined by the fuller harmony, the movement of the soloists relative to each other implies the same tonal quality even though the third note, which would explicitly define that quality, isn't there.