The fact that the food is merely "processed" doesn't make it bad (the classic example is pasteurized milk - it is also a "processed" food; another is properly frozen vegetables).
What CAN make processed food worse than non-processed food is threefour factors:
Addition of certain ingredients that aren't good for you either by their nature (trans-fats) or by the unnatural amount (sodiumsodium).
The additives can be for preservation (e.g. sugar/sodium), or as replacement for easily perishable ingredients (e.g. trans-fats - see Wiki transfats article or http://www.nap.edu/openbook.php?isbn=0309085373&page=423), or to give taste/color (sugar againe.g. MSG - which doesn't seem to have a firm scientific consensus on long term effects but a proven potential for short term ill effects in some people).
A combination of different ingredients may lead to unexpected chemical reactions producing resulting chemicals that may be harmful. A well known example is Sodium Benzoate turning into carcinogenic Benzene in certain conditions.
The quality of ingredients in processed food may not be as high. When you get a hot dog, there's no way for you to examine the meat which went into it for quality as you would the meat you buy from the store. This is because the processing pretty much destroys most indicators of food quality by changing ingredients' texture, mixing them with other ingredients etc... This allows the manufacturer to use lesser-quality ingredient to contain costs (not necessarily pig snouts of urban legend, but just lesser quality cuts etc...).
The processing destroys good/useful/healthy properties of food. E.g. heating destroys vitamins.