In my view, neither are completely clear on their own. "...during the latter part of the 20th century" may sound correct, but how large is a "part"? The first part could be one year, and the following part could be the remaining 99. Even worse, "...during the later part of the 20th century" allows even more ambiguity, essentially meaning nothing, and readers will inevitably misread it as 'latter'.
The first phrasing could be useful with some qualification, for example:
"...during the latter part of the 20th century (namely 1948-1997)"
This description would not be redundant if your goal was to talk about the significance of this period of time in the context of a larger history. For example, that might have been a time of economic boom, whereas the former part was one of bust. I'm not sure if 'namely' is the best word to use in that example, though.