Is the lymphatic system the same as the immune system? Are both of these terms used to describe the same system or are they separate?

Lymphatic system consists of lymph, lymph vessels, lymph nodes and some organs (tonsils, spleen, thymus, vermiform appendix, Peyer's patches).

The diagram of the lymphatic system :

The organs of lymphatic system is the same as the organs of immune system. But these two are different in function.

Lymphatic system has several functions like to drain protein-containing fluid that escapes from the blood capillaries from tissue spaces ; to transport fats from the digestive tract to the blood; to produce lymphocytes and to develop humoral & cellular immunities.
So, the lymphatic system apart from developing immunities, does other things too.

Human body has two major types of Immunities : Innate and Acquired. The humoral and cellular immunities which are developed by lymphatic system is associated with acquired immunity.

The innate immunity is non specific type of immunity and is present by birth. This is not developed by lymphatic system. One example of innate immunity is the presence of HCl (hydrochloric acid) in stomach.

Although these two system share the same organs, functionally they are different.

Here is the organs of the immune system :