Dark roast coffee is roasted longer and at higher temperatures than light roast. What is the difference in taste?
Strength is an interesting wording, but some people use it. Mostly, caramelized or burnt flavors get visible during the roasting process.
As you may see in that answer, Wikipedia enlists roast degrees and their profiles. I would like to summarize them here.
22 °C (72 °F), Green Beans: You may prepare green coffee tea from these. But not the regular coffee.
165 °C (329 °F), Drying Phase: Beans dry out. You may hear the first crack.
196 °C (385 °F), Cinnamon Roast: A very light roast level which is immediately at first crack. Sweetness is underdeveloped, with prominent toasted grain, grassy flavors, and sharp acidity prominent.
205 °C (401 °F), New England Roast: Moderate light brown, but still mottled in appearance. A preferred roast for some specialty roasters, highlights origin characteristics as well as complex acidity.
210 °C (410 °F), American Roast: Medium light brown, developed during first crack. Acidity is slightly muted, but origin character is still preserved.
219 °C (426 °F), City Roast: Medium brown, common for most specialty coffee. Good for tasting origin character, although roast character is noticeable.
225 °C (437 °F), Full City Roast: Medium dark brown with occasional oil sheen, roast character is prominent. At the beginning of second crack.
230 °C (446 °F), Vienna Roast: Moderate dark brown with light surface oil, more bittersweet, caramel flavor, acidity muted. In the middle of second crack. Any origin characteristics have become eclipsed by roast at this level.
240 °C (464 °F), French Roast: Dark brown, shiny with oil, burnt undertones, acidity diminished. At the end of second crack. Roast character is dominant, none of the inherent aroma or flavors of the coffee remain.
245 °C (473 °F), Italian Roast: Nearly black and shiny, burnt tones become more distinct, acidity nearly eliminated, thin body.
Dark roast is stronger taste than light roast, given the same beans.
Bitterness tends to dominate the flavors: Original flavors you can feel on light roast usually disappear with darker roasts.
Dark roast taste is also very stable. It means the flavor remains the same over several roast batches. It is way easier to control flavor of dark roast, then (which is why large chains like Starbucks and McDonald's focus on dark roasts). Light roast are harder to control. The resulting taste depends more on the beans (did they get enough light, water, etc.) themselves, whereas "all beans become the same" with a dark roast.