While the common approach to algebraic groups is via representable functors, it seems that there is no such for differential algebraic groups (defined by differential polynomials). Neither the book by E. Kolchin, nor the texts by Ph. J. Cassidy contain anything like this — they work only with the groups of points over differential fields (and, naturally, don't say the words "group of points").

Concerning difference algebraic groups, i.e. defined by polynomials with some fixed endomorphism (also, I don't like the ambiguity with the notion of "difference algebraic equation"), there is no systematic treatment at all, although some of these groups are intensively studied (twisted groups of Lie type as an example).

So the question is: is there really no modern (scheme-like) exposition of the subject? If so, why?

Differential algebraic groups of finite dimension. Lecture Notes in Mathematics, 1506. Springer-Verlag, Berlin, 1992. – Jim Humphreys Nov 20 '12 at 17:11