Allowable Stress Design is a unique design practice that is adopted by civil engineers while they work on their building projects. It is a practice which entails the designer ensuring that the stresses imposed on the structures owing to the service load don't exceed the elastic limit.

Why is LRFD better than ASD?

Comparing both on the same building design, the general consensus is that LRFD will result in stronger structures for more highly dynamic loads and ASD will result in stronger structures for less variable (more predicable) loads.

What is the difference between strength design and allowable stress design?

The old ASD (allowable stress design) is the same as the new ASD (allowable strength design). The difference is it now has you check the strength (P, M, etc) as opposed to a stress (fb, fc, etc). That is the only difference to the approach, which really isn't a difference at all.