Classification and use of Stocks vs. Broth:
Broths are the result of cooking meat, not just bones. They're generally the result of preparing another item and usually not prepared specifically on their own. The juices poured off from a roasted turkey (after being degreased) would be considered broth. Whole chickens being poached for another preparation would create broth.
Stocks are made from just the bones. They are prepared specifically for use in other recipes (sauces, soups, stews, rice, etc.) Stocks are never salted in their preparation or the finished dish will most likely end up too salty due to reduction that will take place upon further cooking. Note that homemade stock will be often a bit more broth-like than restaurant/commercial stocks, since it's really hard to get all the meat off the bones.
Stocks are usually simmered for a very long time (4-6 hours for chicken & 8-12 for veal/beef) to extract maximum flavor and gelatin from the bones.
Broths aren't usually cooked nearly as long due to the fact that cooking the meat for extended periods (even chicken surrounded by the liquid) will result in tough, flavorless meat.
Consomme: a fortified and clarified stock. The stock is fortified in flavor by the addition of a "raft" which is a combination of lean ground meat (appropriate to the type of stock being used) with brunoise (1/16 inch) mirepoix (carrots, onions, celery), and egg whites. The raft mixture is stirred into the cold stock and as it gently heats, the proteins coagulate forming a "raft" on top of the stock. A small hole is poked in the center (if one hasn't already formed) and as the stock bubbles through the hole it leaches back through the ground meat/egg white raft which filters out impurities to clarify the stock and fortify it with flavor.
Bouillon: French word for broth.
Court Bouillon: sometimes called a "short broth". A poaching liquid usually used for fish that is usually comprised of water, acid (lemon juice, vinegar, wine), parsley stems, bay leaves, peppercorns, and some salt.
When to use Stock vs. Broth:
Use stock when a sauce is to be reduced significantly or when clarity of the final result is preferred.
Broths can be substituted for stock when the body of the liquid or clarity isn't important, and when the liquid will be thickened by addition of a starch.
Skimming is for aesthetic purposes.
The scum is denatured protein, mostly comprising the same proteins that make up egg whites. It is harmless and flavorless, but visually unappealing. Eventually, the foam will break up into microscopic particles and disperse into your stock, leaving it grayish and cloudy. The more vigorously your stock bubbles, the faster this process will occur.
If the grayness or cloudiness bothers you but skimming is not an option for some reason, you can always remove the micro-particulates later through the clarification process used to make consomme.
Best Answer
Brief answer: no, you shouldn't be worried.
Slightly longer answer: you only should be worried if your stock/broth displays characteristics of unskimmed stock (i.e., cloudiness, particles, or odd color) and that bothers you in your particular application for the stock/broth.
Long answer:
There are lots of things that can reduce the amount of apparent foam, including:
With the exception of pre-cooking the meat, most of these won't actually result in less foam/scum in the final product. They just will cause the foam to break up more and be dissolved into the stock/broth so it isn't as noticeable during the initial cooking and can't be skimmed easily. (Boiling the liquid will in fact increase the amount of dissolved particulates, which is why many recipes insist on keeping the heat to a very low simmer.)
If you really want to skim the maximum amount of foam, then start with cold water, heat very slowly, wait to add vegetables and other floating items until after skimming is complete, and never have the temperature above a very slow simmer.
But ultimately the question is whether you should care about removing the foam. There's an old question that addresses this point. Very briefly, there are three main negatives when you don't skim:
If you don't care about these, there's no reason to worry, whether your liquid foams or not. And if you do care about these, you can also clarify your stock afterward (e.g., with egg white).