There isn't any real difference that I can think of except that we would tend to use malnutrition more. The noun "undernourishment" is not used often to my knowledge but we do say that someone is "undernourished" or "malnourished". There may be a slight difference in degree. I would tend to think of someone suffering from malnutrition to be in a worse state than someone who is undernourished.

I'm not sure but I think "undernourishment" means you don't eat enough to live and "malnutrition" means you eat in a wrong way, with abnormal quantity of food components that can make you become ill.

In theory, "malnutrition" means the wrong food and "undernourishment" means too little food. In practice "malnutrition" means the person has so little to eat it is a wonder that they are alive. "undernourishment" is a rare word, and would probably be used in a scientific context: in this case the difference between what the person ate and should eat could be big or small.