Re-entrant tuning as Wiki attempted to explain when used on a uke or tenor banjo makes the chords have "closed-voicings" This opposite to open tuning in fifths for the tenor which makes for "open-voicings". The character of the sound is different. In a band, the closed voicings has a good accompaniment feel especially if the tenor is tuned to GDAE with the re-entrant fourth string an octave higher. In a trad or dixie band, the lower tuning doesn't compete with the high notes of the clarinet and the re-entrant fourth string an octave higher has a fuller and more present sound than the open-voiced standard tenor.

The re-entrant tuning on the banjo and the uke carry better.

As for the use of early lute tuning, the higher string is above the normal E string of the guitar. The Elizabethan 13 string (though some strings are double-coursed) lute tuning is   E-A-D-G-B-C-G. You can approximate the tuning on a six-string as: E-A-D-F#-B-E.

Early music is written better for lute and the later more Spanish harmonic
music is suited to guitar. The early Vihuela (not the Mexican five-string uke) is tuned
usually as E-A-D-F#-B-E and this is how Louis Milan wrote for his pieces in the French
court. This Vihuela is the forerunner of the guitar.

In short, re-entrant tunings give you a set of chords which employ open strings in a different way which create different musical moods. This is certainly illustrated by DADGAD tuning on a guitar.