What is the difference between supplementary angles and a linear pair?

Linear pair is a pair of two supplementary angles. But two supplementary angles might or might not form a linear pair, they just have to "supplement" each other, that is their sum should be ##180^o##.

There are four linear pairs formed by two intersecting lines. Each pair form supplementary angles because their sum is ##180^o##.

There might be two angles that sum up to ##180^o##, but that do not form a linear pair. For example, two angles in a parallelogram that share a common side.