The United States Assistant Secretary of the Navy was (and is) an administrative post designed to give greater civilian oversight into Naval affairs and (in the late 19th and early 20th century) groom future civilian politicians for higher leadership. The Naval Chain of command ran thru the Secretary of the Navy and the President as Commander in Chief. An assistant secretary would mostly just handle tasks delegated to him by the Secretary of the Navy and isn't a very responsible position, although prestigious. Most of Roosevelt's tenure the US faced rather limited naval threats and was at peace except for 1917-18.

In contrast, Churchill served twice as a wartime First Lord of the Admiralty. At the time a First Lord was theoretically at the top of the Naval Chain of command, responsible for setting long term policy and making executive decisions in time of war. The First Lord was a cabinet member accountable to Parliament directly in theory, and in practice the Prime Minister himself. A dominating personality and political talent like Churchill could have very great powers over U.K. naval strategy, procurement, personnel, etc. In every way Churchill's responsibilities in 1911-15 and 1939-40 were vastly greater than FDR's from 1913-1920, and the naval position of his country more perilous.

I don't believe Churchill and FDR had any professional contact in 1913-15 when their positions overlapped. In the 1930s, when FDR became President, and Churchill was only a back bench MP, Churchill would correspond with the President and sign letters "Former Naval Person" to emphasize a possible connection. It's unlikely that Churchill really felt their responsibilities to be equivalent, but was intending to amuse and perhaps flatter FDR by the comparison.