They are different though similar problems. Y2K had more to do with entering dates as two digits. Where as the 2038 problem has more to do with the space that an epoch date takes within data storage. Particularly in strictly typed languages (e.g. C/C++).

Since date functions are usually libraries or built into languages newer versions almost always take this into account, since the problem has been know about for a while. Like Y2K the question becomes what legacy software (and in some cases hardware) is around that will end up breaking.