The study of holomorphic functions and their boundary values is a fundamental part of complex analysis, so it is natural to compare the Bergman and Szego projections associated to a given domain and gauge how closely they are related to each other. In the first part of this talk, I will provide an overview of known results in one and several complex variables for domains with $C^\infty$ smooth boundary. I will then focus on (bounded) simply connected planar domains that are not $C^\infty$ smooth. For such domains with Holder continuous boundary, the difference between these projections gains a derivative in an appropriate range of Sobolev or Lipschitz norms (joint work with Loredana Lanzani).