The mean-tone system, in order to have pleasant-sounding thirds, takes rather the opposite approach from the Pythagorean. It uses the pure major third (Major and Minor Intervals). In this system, the whole tone (or Half Steps and Whole Steps) is considered to be exactly half of the pure major third (this is the "mean", or average, tone, that gives the system its name). A semitone (or Half Steps and Whole Steps) is exactly half of a whole tone.
These smaller intervals all work out well in mean-tone tuning, but the result is a fifth that is noticeably smaller than a pure fifth. And a series of pure thirds will also eventually not line up with pure octaves, so an instrument tuned this way will also have a problem with wolf intervals.
As mentioned above, Pythagorean tuning made sense in medieval times, when music was dominated by fifths. Once the concept of harmony in thirds took hold, thirds became the most important interval; simple perfect fifths were now heard as "austere" and, well, medieval-sounding. So mean-tone tuning was very popular in Europe in the 16th through 18th centuries.
But fifths can't be avoided entirely. A basic major or minor chord, for example, is built of two thirds, but it also has a perfect fifth between its outer two notes (see triads). So even while mean-tone tuning was enjoying great popularity, some composers and musicians were searching for other solutions.