It seems to me that "exact" relates to exact differential equation. So, why are they called exact?

According to Hans Samelson's historical note "Differential Forms, the Early Days", both notions were introduced in Les Méthodes nouvelles de la Mécanique Céleste by Poincaré (vol. 3, GauthierVillars, Paris, 1899, pp. 915). Samelson notes
Apparently, it had taken some time for the terminology to stabilize as, for instance, Goursat used the term "exacte" in his book for a form that today one calls closed (E. Goursat, Leçons sur le problème de Pfaff, Hermann, Paris, 1922). 


This is not an answer to the original question but closely related, so I leave this post here for the interested reader There are many variants of the story but here's what I found by googling which is as good an account as any other:
Copied from The Exact Answer to a Question of Shields by Donald Sarason, Mathematical Intelligencer, Vol. 12, No. 2, 1990. 


Well, the notion of exactness lies in an algebraic background. Given a sequence of groups or RModules and morphisms given by arrows in the following way: ... > A_n f_n> A_n+1 f_n+1> A_n+2 > ... we call it exact whenever Im f_n = Ker f_n+1. Normally forms in Global Analysis are related to De Rham Kohomology which is precisely the quotient of such sequences for certain RModules (or CModules). The de Rham Cohomology group of certain order is trivial whenever the short sequence is exact (exactness in the 3 modules involved), this occurs exactly when all the closed forms are exact. About "close" I dont have an answer although I may think of some reasons I prefer not to comment hehe. 


Because of homological meaning and the relation with simplicial cohomologies? 

