They differ in number and type differentiated cell types they can become.

Differentiation Capacity

While embryonic stem cells can become all cell types of the body because they are pluripotent, adult stem cells are  limited to differentiating into their tissue of origin.

Growth

Embryonic stem cells can be grown relatively easily in culture. Adult stem cells are rare  and isolating these cells from an adult tissue is difficult. Moreover ways to increase their population in cell culture is yet to be found.

This is an important difference, as large numbers of cells are needed for stem cell replacement therapies.

Rejection Chances

Embryonic and adult stem cells may differ in the likelihood of being rejected after transplantation. Though not supported by any scientific evidence, adult stem cells, and tissues derived from them, are currently believed less likely to initiate rejection after transplantation. This is because a patient’s own cells could be expanded in culture, coaxed into assuming a specific cell type  and then reintroduced into the patient.

The use of adult stem cells and tissues derived from the patient’s own adult stem cells would mean that the cells are less likely to be rejected by the immune system.

This represents a significant advantage, as immune rejection can be circumvented only by continuous administration of immunosuppressive drugs, and the drugs themselves may cause seroius side effects.