Not a teacher.
As I understand it, a "specimen" is whole and a "sample" is a part. You draw a blood sample. It's not all of your blood, but a representative part of the whole.
A "specimen" is like taking one or more whole individuals of a living creature. Usually bacteria or that sort of thing. A specimen is one of a kind of thing. A sample is part of a larger volume or collection of a thing.
You don't take urine specimens and they don't take throat samples if they think you have strep throat.