What are the fundamental differences between a cell membrane and a cell wall?

Both encase different type of cells however their compositions are distinct and because of this they have different functions and distributions across cell types. 
Structure and compositionCell membranes are very flexible membranes made up of a lipid bilayer embedded with proteins and carbohydrates. They are also very fluid in that they are in constant movement around the cell and may have different quantities of proteins and carbohydrates depending on the cell type and throughout the course of its life depending on the needs of the cell. In contrast, cell walls are very rigid and provide significant structure and shape for cells that have it. Their composition can be different depending on the type of cell such as cellulose for plants, chitin for fungi and peptidoglycan for bacteria. 
FunctionThe basic function of the cell membrane is for the cell’s protection; however, this goes hand in hand with its ability to separate the intracellular and extracellular space and, just as important, letting important things in and out such as nutrients and waste products. They are semi-permeable which means they are very selective in letting things in or out. The cell membrane is an organelle and is metabolically active. The cell wall is very different in that it functions mainly for the cell’s structural integrity and as its filtration system. It is completely permeable such that it is able to let small molecules through and because of this, cell walls are also act as pressure vessels meaning that they also prevent the cells from bursting such as when water enters. Despite their rigidity, they are also elastic and this contributes to their strength in motion. The cell wall is also an organelle but it is metabolically inactive. 
DistributionCell membranes exist in all types of cells however cell walls only exist in bacteria, plants, fungi and algae. In a diagram, cell membranes are the inner lining and, if present, the cell walls are the outer line. 
Fun factsNeuron membranes