Cmaj7 and C7 under a hand
Cmaj7
Strummed from the lowest string to the highest, this grip sounds C, G, B, E, G and C, which are the degrees 1 5 7 3 5 1 rather than the 1 3 5 7 of the written chord. The root is the lowest note, so this grip is in root position and sounds the way the written chord suggests. With C in the bass and six strings sounding four notes, the 1 and 5 appear more than once.
Strummed from the lowest string to the highest, this grip sounds E, C, E, G, B and E, which are the degrees 3 1 3 5 7 3 rather than the 1 3 5 7 of the written chord. The lowest note is the 3 rather than the root, so this grip is in first inversion, written Cmaj7/E. It has the same notes as the root-position grip, but with E in the bass it sounds different. With E in the bass and six strings sounding four notes, the 3 appears more than once.
Strummed from the lowest string to the highest, this grip sounds G, C, E, B and E, which are the degrees 5 1 3 7 3 rather than the 1 3 5 7 of the written chord. The lowest note is the 5 rather than the root, so this grip is in second inversion, written Cmaj7/G. It has the same notes as the root-position grip, but with G in the bass it sounds different. With G in the bass and five strings sounding four notes, the 3 appears more than once.
Strummed from the lowest string to the highest, this grip sounds B, E, G, C, E and B, which are the degrees 7 3 5 1 3 7 rather than the 1 3 5 7 of the written chord. The lowest note is the 7 rather than the root, so this grip is in third inversion, written Cmaj7/B. It has the same notes as the root-position grip, but with B in the bass it sounds different. With B in the bass and six strings sounding four notes, the 3 and 7 appear more than once.
C7
Strummed from the lowest string to the highest, this grip sounds C, G, B♭, E, G and C, which are the degrees 1 5 ♭7 3 5 1 rather than the 1 3 5 ♭7 of the written chord. The root is the lowest note, so this grip is in root position and sounds the way the written chord suggests. With C in the bass and six strings sounding four notes, the 1 and 5 appear more than once.
Strummed from the lowest string to the highest, this grip sounds E, B♭, G, C and E, which are the degrees 3 ♭7 5 1 3 rather than the 1 3 5 ♭7 of the written chord. The lowest note is the 3 rather than the root, so this grip is in first inversion, written C7/E. It has the same notes as the root-position grip, but with E in the bass it sounds different. With E in the bass and five strings sounding four notes, the 3 appears more than once.
Strummed from the lowest string to the highest, this grip sounds G, C, G, B♭, E and G, which are the degrees 5 1 5 ♭7 3 5 rather than the 1 3 5 ♭7 of the written chord. The lowest note is the 5 rather than the root, so this grip is in second inversion, written C7/G. It has the same notes as the root-position grip, but with G in the bass it sounds different. With G in the bass and six strings sounding four notes, the 5 appears more than once.
Strummed from the lowest string to the highest, this grip sounds B♭, E, G, C, E and B♭, which are the degrees ♭7 3 5 1 3 ♭7 rather than the 1 3 5 ♭7 of the written chord. The lowest note is the ♭7 rather than the root, so this grip is in third inversion, written C7/B♭. It has the same notes as the root-position grip, but with B♭ in the bass it sounds different. With B♭ in the bass and six strings sounding four notes, the 3 and ♭7 appear more than once.
Four shapes for Cmaj7 on guitar: Cmaj7, Cmaj7/E, Cmaj7/G and Cmaj7/B. Four shapes for C7 on guitar: C7, C7/E, C7/G and C7/B♭.
These two chords differ by one note. If you find a grip where only one finger moves between them, that finger is playing the note that makes the difference.
If you add one more third on top of a triad, you have a seventh chord, with four notes rather than three. On C that gives C, E, G and B, a chord called Cmaj7.
Lower that B by a semitone to B♭ and you have C7, which behaves very differently. Cmaj7 sounds settled. C7 contains a tritone, the interval of six semitones between E and B♭, and a tritone sounds restless. It creates a pull towards another chord, usually the one a fourth above, which for C7 is F. That single semitone drives the chord changes in blues and in a great many jazz standards.
In open position, both chords are one finger away from the C chord you already know.