Here's the advantages/disadvantages of both.
Eating before you run
If you eat about three hours before you do a 10k training run, you'll have broken down all your sugars, proteins and fats into soluble products including glucose, sucrose, amino acids, triglycerides etc etc. Your glucose will be converted to glycogen and you'll have a big store of energy for running so you can go for a long run and perform at your best. However, if your glycogen runs out, amino acids will instead of undergoing anabolism to form proteins will be undergo gluconeogenesis and will not build proteins and so you will not gain any muscle content post training. You may be liable to digestive discomfort if you don't give yourself enough time to digest.
Eating after you run
Your muscles' protein content will diminish ever so slightly due to lack of readily available sugars after all of your glycogen has been used up. Your glycogen levels will be much lower because you won't have taken on board any extra sugars.
Personally, I eat four hours before running and will have a light snack two hours after running.
