Donald Nicol has actually written a book on the family, and another one on John Kantakouzenos. They were an immensly wealthy, landed family with a sense of entitlement - something which is said to have influenced John Kantakouzenos' politics - who rose during the Komnenian and Palaiologan period. The Cantacuzino family might be related to them, but their connection is not clearly established.

You can't research you family's history by asking what was going on in the Byzantine Empire. You've been told that multiple times from the looks of things. I did a basic search for the person you asked about and found some interesting results:

• In 1937 Michael Cantacuzene-Speransky is listed as stateless. He's well educated, from St. Petersburg, and has likely ties to the Polish person next to him.
• In 1947 he's still stateless but married now to an American citizen.

I suggest you contact someone who specializes in Eastern European genealogy if you want to dig further back. You might have some big hurdles ahead of you with the history of the regions your surname is associated with.