I have a question under this subject that I've been wrestling with for years!

I began my main website in 1999. Over the years I've changed hosts and found that some hosts didn't allow me to change the .htaccess file, so I had to create a index.htm file instead of a index.html file for the home page.

I noticed back in 2001 that different browsers would display different pages as the home page htm VS html.
Different search engines would link to one or the other as the home page.

Of course, years later I now have both the index.htm and index.html page for my home page, both with the same content.
Should I use a 301 redirect for one of the index pages instead of continuing to duplicate the content.
If yes, which one do I delete and redirect to the other?

Thanks for your consideration and replies.