Double tapers are easier to mend and to roll cast with, I use them for small river fishing I think Double Tapers generally give you more control. However the down side is there are not as easy to cast and they won't give you as much distance as a Forward weight line.
A double taper has the exact same (relatively long) taper on both ends and a long belly in the middle. It can be reversed if the front taper "wears out" (I don't know what that means), but that is not its only function, advantage, or disadvantage. The more common (these days) "foward taper" has a tapered section in front, a short belly, and a very short rear taper, followed by a long section (the majority of the line)of relatively narrow, level line, the "running" line.
On medium-distance casts, the two different types will perform about the same, all else being equal. Double tapers tend to perform a little better on very short casts, particularly if you're trying to turn over very long, fine leaders, and they mend a little better because of the big belly. On long casts, they have a small advantage that quickly becomes a disadvantage. Because of the long taper, the line is always turning lighter line over heavier. Theoretically, this makes for better, if rather open loops. As casts get longer, however, double tapers are harder to throw. They require you to carry almost the whole cast in the air, because they don't "shoot" very well, because the belly is heavier than the taper and because they don't make very tight loops. Carrying all that line in the air demands precise timing, not to mention alot of backcast-room.
A forward taper has all the taper --and weight-- in the first 30 or so feet. On casts longer the 30', they have a slight disadvantage that quickly becomes an advantage. As you start carrying the much thinner running line in the air, it has a tendency to "hinge" the loop and break it down. Mends are tougher because you have the very light running line between you and the heavier tepered section. On the other hand, FTs throw much tighter loops. Tight loops are better in the wind, and once mastered, will let you shoot the entire length of thinner, lighter running line. You could throw a 90' cast, and only have to carry 30' in the air (I only know of about 2 or 3 chaps who can actually do this, but you get the picture).
If you need to cast far or in windy conditions a forward taper is the way to go. Their disadvantages for short and medium casts are pretty small, and most guys use FTs for everything. Double tapers are probably considered a little old fashioned these days, but they do still have some applications. For very technical trout fishing, particularly with lighter rods (4 and below), they work great. Some anglers like heavy DTs for spey rods too, because the long rod and complicated cast requires a long-bellied line. For the same reason, DTs will generally make longer roll casts.
DT's, because of the long thick belly, take up more room on a reel than the same weight FT line,leaving less room for backing, which tends to make them a little less practical, unless you're going after small fish, using a very light line, or a very big reel (like a spey reel).