From the theory of Hilbert bases and separability considerations one can easily derive a Hilbert space isomorphism between $H^1(0,1)$ and $L^2(0,1)$. I am convinced that such an isomorphism is useless: I know explicit examples of $L^2(0,1)$ functions whose distributional derivative is not $L^2(0,1)$ so in pratice (e.g. when inspecting the regularity of a solution of a PDE) I do not wish to identify these two spaces. So my question is:
Why don't I like this isomorphism? What should a "good" isomorphism satisfy?
You can find isomorphism $A: H^1 \leftrightarrow L^2$ preserves addition, multiplication by scalar and inner product (and so norm, topology, convergence, etc.). But there is a lot of additional structures on both $H^1$ and $L^2$ that are not preserved. One example you already gave: you have a differentiation operator $D: H^1 \to L^2$, which is easy to define, but operator $X = DA^{-1}$ that corresponds to it will be very strange.