While very similar in spelling and somewhat similar in meaning, clench and clinch do mean different things and are used in different situations.
Clench: contraction or tightening of a body part, such as jaws, hand (into a fist), and sphincter
Clinch: an embrace (the two illicit lovers were found in a clinch)
or, in sports when I was much younger, winning some aspect of play, such as a season, before that aspect of play was over
Formerly, "clinch" was used mostly near the ends of regular seasons (such as in baseball, American football) but before the season was over in situations that a team's lead was too large to overcome in the length of season remaining. Once the season was completed, "clinch" was no longer used. Instead, the team was said to have "won the league (or division or whatever)."
Now, however, it's become a synonym of "won."