Degenerate denotes a decline that is mental or moral, for example "The debate degenerated into a shouting match" = the debate was once civil but it became hostile Deteriorate is more physical, for example "his health was deteriorating" = his health was getting worse

As a noun degenerate can refer to a corrupt person. Deteriorate is not used in this way. Otherwise they are both very similar in that they both mean a decline from a once better state. Or an ongoing/continued decline depending.