I would add an important distinction of LC singularities to Karl's excellent answer above: DLT (and in particular KLT) singularities are rational, while LC are not. This already appears when the boundary is empty. LC singularities are not even necessarily Cohen-Macaulay, such as for example a cone over an abelian variety of dimension at least 2. I think that this is the main reason for LC pairs being so much harder. For LC centers, it's also a good idea to look at Ambro's work on quasi-log varieties. That seems to be a very effective way of handling LC centers. This is evidenced by this recent paper in JAMS and Kollár's local Kawamata-Viehweg vanishing result (on the arXiv).
Another comment on LC centers: this is probably obvious for most people, but the point of an LC-center is that that's why a pair is not KLT. They are also called "non-klt" centers. I believe Christopher Hacon prefers this terminology. As far as I understand, LC centers are similar in spirit to associated primes. The simplest thing about a module is its support and in particular its irreducible components. However, considering associated primes gives a better understanding as there are components that kind of should be considered part of the support, but they are embedded, sort of "shadowed" by other components. The union of LC centers is exactly the non-klt locus, but just like with associated primes, there may be some that are embedded, so knowing the LC centers gives more refined information about the failure of the pair to be klt.