My sense is that it'll be a lot worse, Y2K was only a data format issue, whereas 2038 issue has to do more with underlying hardware. It really all depends, and we'll see. Certainly, many old software will stop working in 2038.
I'd say both are a data format issue, y2k was usually at a higher level and occured in custom data formats of individual software, while 2038 is in the OS and basic libraries, or even in hardware.
I do think though there were some bioses that messed it up too, so that's rather low level too.
