Both are transitive verbs that translate as "to force" in English.
Zwingen and zwängen come from the same Germanic root that meant "to compress a body by force". This physical origin is still visible especially in zwängen.
Zwängen means to force a body through a somewhat narrow gap or hole. It either takes a reflexive object or an object that is not a living being.
Ich zwänge mich in die U-Bahn.
Ich zwänge das Buch in die viel zu kleine Tasche.
But you cannot really zwängen another person as that would imply that you physically "compress" that person.
Zwingen on the other hand is mainly abstract. "Compressing" is not the point anymore, just "strong, irresistible pushing" if you will. It means to force in all kinds of ways and it usually takes a person as a direct object.
Ich zwinge mich, den Computer auszumachen.
Ich zwinge dich zu nichts.
Mind you, that zwingen is a little more narrow than force in that it kind of implies some sort of consequence. You cannot really zwingen open a door because you have nothing to pressure the door with. You cannot threaten the door with consequences, hence you can't zwingen it. Technically, zwingen can thing as an object but at least to me that results some kind of personalization of the object.
Ich zwinge den Router sich zu synchronisieren.