Why do we need admissible isomorphisms for differential Galois theory?

I don't think I have seen the terminology "admissible isomorphism" being used in differential Galois theory, except for Kaplansky's book. I guess in E. Kolchin's work everything is assumed to lie in a universal differential extension, and therefore he never makes the distinction.

Your observation about Picard-Vessiot extensions is right, and I don't think one needs the notion of admissibility to develop ordinary Picard-Vessiot theory, which is a theory based on the "equation" approach. In fact the efforts done at the time were focused on developing a Galois theory of differential fields that wasn't necessarily associated to differential equations (but it had to be a generalization of PV of course). However, many problems arise when one takes this "extension" approach, in fact finding the right notion of a normal extension is not easy. Classically a field extension $M$ over $K$ is normal if every isomorphism into some extension field of $M$ is an automorphism. However the equivalent statement for differential algebra implies that $M$ is algebraic over $K$ and that is too strong (in fact this is one of the main reasons why one has to allow admissible isomorphisms). Here are two early approaches to normality:

$M$ is weakly normal if $K$ is the fixed field of the set of all differential automorphisms of $M$ over $K$.

Apparently this definition wasn't very fruitful, and not much could be proven. The next step was the following definition:

$M$ is normal over $K$ if it is weakly normal over all intermediate differential fields.

This wasn't bad and Kolchin could prove that the map $Lto Gal(M/L)$ where $Ksubset Lsubset M$ bijects onto a certain subset of subgroups of $Gal (M/K)$. However the characterization of these subsets was an open question (Kolchin referred to it as a blemish). The property he was missing was already there in the theory of equations, as the existence of a superposition formula (that every solution is some differential rational function of the fundamental solutions and some constants). The relevant section in Kaplansky's book is sec 21. Now an admissible isomorphism of $M$ over $K$ is a differential isomorphism, fixing $K$ element wise, of $M$ onto a subfield of a given larger differential field $N$. Thus, an admissible isomorphism $sigma$ let's you consider the compositum $Mcdot sigma(M)$ which is crucial to translating a superposition principle to field extensions. Indeed, if one denotes $C(sigma)$ to be the field of constants of $Mcdot sigma(M)$, then Kolchin defined an admissible isomorphism $sigma$ to be strong if it is the identity on the field of constants of $M$ and satisfies
$$Mcdot C(sigma)=Mcdot sigma(M)=sigma (M)cdot C(sigma)$$

This was the right interpretation of what was happening in the PV case and so a strongly normal extension $M$ over $K$ was defined as an extension where $M$ is finitely generated over $K$ as a differentiable field, and every admissible isomorphism of $M$ over $K$ is strong. Now the theory became more complete. $Gal(M/K)$ may be identified with an algebraic group and there is a bijection between the intermediate fields and its closed subgroups. Now this incorporates finite normal extensions (when $Gal(M/K)$ is finite), Picard-Vessiot extensions (when $Gal(M/K)$ is linear) or Weierstrass extensions (when $Gal(M/K)$ is isomorphic to an elliptic curve).

For a better exposition of this, see if you can find "Algebraic Groups and Galois Theory in the Work of Ellis R. Kolchin" by Armand Borel.