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Stock is a richer, heartier liquid made from simmering bones and vegetables (often browned in the oven before simmering to enrich the stock even more and give it a lovely deep color). Stock should have a good body--meaning, a high gelatin content (if your stock gelatinizes when chilled, you have achieved a high gelatin content).
Broths tend to be a bit weaker and less gelatinous. Sometimes they are made with meat and no bones (thus the substantially lower gelatin content, much of which comes from bones), but some broths do contain bones.
Essentially, it's a matter of degree--stocks are stronger and richer; broths are lighter. I'm sure there are more technical aspects of the two that I'm forgetting here, but I'm sure other knowledgeable Food52ers can further elucidate this oft-confusing dichotomy.
pierino is a trusted source on General Cooking and Tough Love.
Really there is no difference. But as petitbleu noted you should be aiming for that jelly-like aspect after you skimmed and refrigerated. Even after straining there is still likely to be a fat cap after the refrigeration. Easy to remove e voila!
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