Let’s see if I can give it a shot.
Bourbon is a US-made whiskey made primarily from corn, with significant amounts of malted barley and rye and/or wheat, and usually distilled in a column still. It’s then aged in a new charred oak barrel for at least a year in relatively hot conditions (relative to Scotland, anyway), although most commercial specimens are closer to 4.
Scotch is a whiskey made in Scotland primarily of malted barley and generally aged in used barrels of various sorts under cool conditions for a long time. Scotches can be made using peat fires for drying the malt, which gives it a pronounced smoky/medicinal aroma and flavor, but doesn’t have to be. Scotches can be “single malt”, meaning that a single fermentation was distilled, bottled and aged, or they can be a “blended malt”, which means that a bunch of single malts are blended together to achieve a certain flavor profile, or they can be “blended” which means that various single malts can be blended with cheaper and less flavorful grain whiskies, which are made from other grains, and usually with a column still, rather than the traditional pot still used for the malt whisky, which gives it more character.
So what this all means is that the Scotch whiskey tends to have more of the character of the original ingredients, and less of the actual wood flavor from the barrel. This is due to the fact that Bourbon uses new barrels and uses pot stills, and Scotch tends toward pot stills and used barrels, with the pot stills being a less “clean” distillate with more character, and the new barrels imparting far more flavor. (as an aside, used Bourbon barrels are commonly used to age Scotch).
In practical terms, this means that Bourbon has a pronounced wood-derived flavor, which is a sort of vanilla-like sweetness, while Scotch has a fairly defined “malt whisky” flavor that’s hard to describe. That’s in broad strokes; there’s a definite difference between sub-styles within the categories; wheated Bourbon (Maker’s Mark, Weller) is somewhat smoother and less flavorful than high-rye Bourbon(Bulleit, Old Grand-Dad), which has a spicy sort of taste to it.
I’ve actually had (at a Tales of the Cocktail seminar) a sample of Glenmorangie that was aged as if it was Bourbon, for 4 years in a new charred oak barrel, as well as the white dog itself (the unaged distillate), and the same thing, only as finished Scotch. The Bourbon-style aging was very prominent- it was, for lack of a better description, much like a funky Bourbon. The white dog itself was ghastly; it tasted like scotch, but with a dollop of gasoline and something else nasty thrown in. I think the aging is primarily to knock those rough edges off. I haven’t had Bourbon white dog, but I suspect it’s a lot smoother of a spirit, considering that they’re actually selling it at retail now.