If we stick to the linear version and discrete versions of filter banks and wavelets, filter banks represent the generic tool, and wavelets can be implemented as a specific instance of iterated $2$-band filter banks satisfying some additional properties, namely that low-pass spaces are embedded dyadically.
In other words: get a single-level $2$-band perfect reconstruction filter-bank, add some regularity so that iterations converge toward something legit, and you have a concrete Discrete Wavelet Transformation.
Kernel types have almost nothing to do with it, from a theoretical perspective. Kernels are chosen with respect to, mainly, implementation (fast or not) or signal properties (regularity).
That was the simple answer. If you want to dig deeper into it, you can look at continuous wavelets (that remain filter banks, as the admissibility condition impose a wavelet to be, somehow, a bandpass), nonlinear wavelets, morphological or hybrid filter banks, etc.
When I was a student, I was teached filter banks by Maurice Bellanger, one (the?) father of polyphase decomposition (Digital filtering by polyphase network: Application to sample-rate alteration and filter banks, 1976). As a young person, I just heard about wavelets, and asked him about it. His answer was: "they are just some special filter banks".