I found this quite puzzling but thought I might as well ask.

I had some difficulities finding out the years of some banknotes that I have simply because there aren't any years printed on them. On the other hand, almost all modern coins do have a date on it. Why?

Is it because it's more expensive to replace a printing plate compared to a die used to strike coins?

When they do have dates, it is not always indicative of when the note was actually printed. For instance some of the last 1935 series Silver Certificates in the USA were being printed into the early 1960's. Other times notes were printed with a date of 1934, in 1944, but released into circulation up to 1969.

The USSR printed notes dated 1961 until 1991. Russian notes are still dated 1997, but that reflects the new ruble issue.

Ukrainian hryvnia notes usually have the year the note was printed though so it is possible to get a 1 Hryven note dated 2005, 2006 etc.