$C^1$ elements are mostly a historic relic. In the finite element method, the traditional view is that the best methods are "conforming", i.e., methods where the finite element space $V_h$ is a subspace of the space $V$ in which the solution lies. For second-order elliptic equations, $V=H^1$ and functions that are continuous and piecewise polynomial (but not necessarily continuously differentiable) are a subspace of $V$.
But that is not the case for fourth-order equations such as the biharmonic equation. There, $V=H^2$ which contains only functions that are continuous differentiable (i.e., $C^1$). So in that case, the usual Lagrange elements are not a subspace of $V$ and it is not clear how to implement the bilinear form with these elements. So people, going back to the 1960s, developed elements that are $C^1$ and for which consequently $V_h \subset V$. This works, but the elements are quite difficult to implement for non-conforming meshes and they just don't quite fit into the systematic view we have of elements today, the paper by Kirby and Mitchell mentioned in one of the other comments notwithstanding.
But starting in the 1990s, we learned how to use non-conforming elements more efficiently -- first in the form of discontinuous Galerkin methods for elliptic equations and then also how to use Lagrange elements for biharmonic equations. I would specifically refer you to the 2005 paper by Sue Brenner and Sung on the $C^0$ Interior Penalty ("C0IP") method for biharmonic problems that is also used in the step-47 tutorial program of deal.II and that shows how relatively easy it is to solve these kinds of problems with just the usual elements. (Disclaimer: I'm one of the authors of deal.II and of step-47 in particular.)
Now, it is true that the paper by Kirby and Mitchell shows that the $C^1$ elements have advantages regarding condition numbers and solver speeds. But at least in my opinion, I don't think this outweighs the very substantial pain to implementing them on unstructured meshes and meshes that potentially contain hanging nodes. I had a long discussion with Rob Kirby about that paper at some point and have to admit that he's one of my heros for undertaking this kind of project -- he's the only person I know who had the gumption to implement $C^1$ elements in the last 20 years, and I think I know a substantial fraction of the people who implement finite elements :-)