What is the difference between a Branch office and a subsidiary?

A branch office is a different location for the same company
A subsidiary is a smaller company owned and controlled by the major company

A subsidiary is a different company. A branch is the same company.

A Branch is the same comapny name but at another location.
A Subsidiary is another name with new product line or business line - it could be in the same office or in the same branch or at any other location.

the branch office is a BRANCH OFFICE
and subsidiary is a SUBSIDIARY

HUGE - subsidiary is a different legal entity

usually, a subsidiary is a 'complete' and 'seperate' legal entity owned by another legal entity (corporation, partnership, etc) while a branch office does not have a seperate legal standing, it is just a location where an entity conducts business.
Subsidiaries are often previously existing legal entities that are aquired via purchase or merger; some subsidiaries are chartered to meet specific regulatory requirements or to insulate corporate liability pertaining to specific business activities.