Do fresh herbs have properties that dried herbs do not?

The answer is the same for herbs as you got for chili - you lose vitamins (not that you will get much from the amount of herbs usually used in cooking anyway) and some volatile compounds, which are the bits that make the flavors. In general you can expect to use about 2-3 times as much fresh herb as you would dried to get the same level of flavoring in a dish. Note that this amount of dried herb is equivalent to a larger amount of fresh herb as dried has shrunk as it lost water.

The degree to which the herb loses flavor when dried varies depending on the species of herb and most likely is a function of which aromatic compounds are present (i.e. how volatile they are) and how easy it is for them to escape the leaf (toughness of leaf). Some herbs, like sage and oregano, hold their flavor fairly well, while others, like basil and cilantro/coriander, lose their flavor very rapidly and should only be used fresh.

In my experience, if you are only able to get dried herbs, only get those that are whole leaf or crumbled. Do not get the powder ones, as these lose flavor very rapidly, indeed by the time they get to the store they are often almost flavorless. The same principle applies to spices too - whole is better than ground for storage and retention of flavor.

There are some herbs, like bay-leaf, that are best dried as it concentrates the flavor, but these are a rarity. This doesn't appear to be the case for spices, which are generally dried before use, but this may have something to do with many of the spices only growing in tropical areas and needing drying for storage while shipping rather than fresh spices not being good for cooking.