Citing your source would help to answer.
In order to test an elastic modulus, you need to apply some non-zero strain. If the strain is infinitesimal, then compression and stretch modulus will be equal. But if it is finite, then there can be a difference due to the structure of the material.
In bone, I believe porosity will provide most of the dissymetry between compression and extension: at small scales, the material can rearrange using the pores without much deformation. Whereas in extension, there has to be some deformation of the osteons.
In other living tissue, the difference can be due to filamentous proteins: if you compress them, they buckle (form loops) without much resistance. If you exert extensive stress, once they are stretched tight they will resist any further extension. (Think of what a piece of string would do, it's very similar)