Is the palate expander with the key and without the key the same thing?

Yes: If it is called a palatial expander, it should be doing the same function which is to widen the upper arch. There are several types or designs. The expander with the key being the most common and traditional.

Control expansion: Palate expander with the key is an active appliance, in which the key control the expansion. Palate expander without a key usually using a spring or opening a loop bend in the transpalatal arch wire to engage the expansion. Thus a key provides more control both in the force and the amount of expansion. Expansion stops as the use of key discontinue. Force of the spring cannot be stopped.

Sort of: There are literally multiple dozens of different types of expanders. Some are designed to expand death. Others are designed to widen the transverse dimension of your palate. Talk to your orthodontic specialist about why he/she chose the particular expander that he recommended. The "medicine" chosen depends on the nuances of the problem to be treated.