From the latin limitrŏphus, bordering is something close, border or who is confined. The concept is linked to the notion of limit (a real or imaginary line that separates two territories, countries or land).
An administrative political division has several distinct regions by limits imposed by man. Boundaries are boundaries that mark the divisions. Adjacent regions are adjacent (they have common limits).
For example, the Brazil, the Uruguay, the Chile, the Bolivia and the Paraguay are adjacent to the Argentina countries given that this country shares borders with such neighbouring nations. However, the Argentina is not adjacent to the Ecuador, Colombia or any other country part of those mentioned above.
The Russia and China are the countries bordering the largest number of nations: each has 14 neighbouring countries. It is worth mentioning that the bordering adjective can also refer to other entities beyond the countries, such as the provinces, cities, etc.
The term is also used to refer to anything that limits. Thus, one can speak of boundary or border conflict conflict: "the Colombia and the Ecuador are involved in a border dispute for decades.
Boundary, the other hand, as a synonym for "bordering", can be used to refer to something that is close to another, be it physically or symbolically: "singer surprised the audience with a boundary declaration of discrimination", "I don't want anything that is adjacent to smuggling in this city.