Many stores only offer LME in fixed quantities of 3.3 lbs or 6 lbs. Since it's impractical to use only part of a can/jug of LME, DME is frequently used to make up the difference in fermentables between what a can or two of LME provide, and what a style calls for. DME is most commonly sold in 3-lb bags, but if you re-seal them properly, the remainder can be stored for several months.
LME tends to be cheaper, but stales more quickly, and scorches easily if you aren't careful.
DME tends to be more expensive and tends to clump, but it is more stable and doesn't scorch as easily. It is also more concentrated (i.e. 1 lb of DME is equivalent to about 1.25 lbs of LME).
I like LME but I use DME in almost every batch as described above. I buy strictly in cans, NEVER in bulk, from a place where I know they get good product turnover. I think in part the spotty reputation for LME comes from the practice of selling from bulk containers. I've never had mold or off flavors. My only issue with DME is minor - if your pot is boiling away, the steam can cause it to clump up on the bag from which it is being poured.