Are the Corps of Engineers and Engineer Branch one in the same?

So - good question! The name of the branch for all Army engineer officers is the Corps of Engineers, but USACE, or the US Army Corps of Engineers is a unique hybrid government/military organization. Both tactical Army engineer units as well as USACE are led by Army engineer officers - a unique opportunity for those branched engineer. USACE is mostly made up of federal civilian workers: engineers, scientists, administrators and even park rangers. If you eventually branch as an engineer officer, you will start out in the "regular Army," National Guard or Reserves, but after initial assignment(s) as an LT, you will also have access to serving in the "civilian" Corps as an engineer officer at a USACE District. USACE have federal responsibility for all inland waterways in the United States, as well as supervision of major military construction projects such as barracks, hospitals, other large projects - basically anything appropriated by Congress. They also provide specialized engineering expertise to the Army - including downrange. Some engineer officers serve their entire careers in the regular forces, but some seek assignments in USACE along the way. Good Luck!!

No. USACE is an entire entity all its own

No they are not. Way different. Engineer Branch is military and Corps of Engineers is almost fully civilian. With two completely different missions.