Well, "as of" is more like "since" or "starting on this day" where the focus is more on what happens after that day. In your example, it's talking about a program or policy that will start on January 20 and will continue on indefinitely. The focus is on the future as defined by that new policy or program. "On January 20, we will begin the new policy" focuses on the beginning of the policy and what happens on January 20. That the new policy will presumably continue is more of an afterthought.
how different, in this case, is 'as of' from 'on'?
But, on January 20, we...
"as of" is a term I never use. It's not necessary.
As you indicate, it can mean "since".
On the other hand, "As of the 25th we had raised $3,000" means on the 25th we had $3,000. (Here it means "at").
Hence, to different people and in different circumstances, it can mean "up until" and "since", which renders it very ambiguous at times.