Tubular tires - also known as sew-up tires - are a combination of tire and tube all in one. The tube is sewed inside the tire casing, hence the term sew-up tire. Tubular tires are affixed to a bike rim with glue.
Tubeless tires are a style of tire which interfaces directly with a special type of bike rim - a tubeless rim - to create a seal, and no inner tube is required. There is a liquid sealant which is used to create a strong juncture between the tubeless tire and the rim.
So let's compare both types of tires to the very common clincher tire style. Tubular tires are noted for being lighter in weight - and thus contribute less to rotational weight - and having a distinctly good ride quality. Almost all pro bike racers ride on tubular tires. Before the 1970's, tubular tires were the norm, but then clinchers became more popular. Current clincher tires rival tubular tires in terms of ride quality and weight. Another reason that weight comes into play, besides the tires themselves, is that a tubular compatible bike rim has less metal (or carbon) at the outside of the rim compared to a similar clincher rim. That's because a clincher rim needs a "lip" on both sides for the tire to "clinch" to. How do you change a tubular tire if you flat? You carry a spare tire - that's right, a while tire/tube combo. You put the replacement tubular tire on the bike wheel and continue your ride, realizing that the glue bond will not be nearly as strong as with the original tire. Actually, in some sense, there is no "real" bond, since the tubular tire glue is meant for one application. So you will need to ride a bit carefully after swapping out a tubular tire, particularly on turns and at high speed going downhill. To fix a tubular tire, you cut it open with scissors (or an exacto knife) around the area that the puncture has occurred, patch the tube inside, then sew up the tire again and replace the inside tire tape.
Tubeless tires offer some similar advantages as tubular tires. They are lighter in weight, in this case, because there is no tube. If you flat, a tubeless tire won't separate from the rim. To fix a flat, you inject new sealant or install a spare inner tube, which you will need to carry. Less rotational weight means less power to get tubeless wheels spinning which means you go faster with the same amount of power generation.