For awhile there was an argument in the scientific community whether this phenomenon was due to a smell disorder or a urine problem...
EVERYBODY's urine smells... it's just that half the population can't smell it.
It's not a nose problem... it's a urine problem due to an inability to digest (or resorb) sulfur-containg compounds found within asparagus after breakdown. In fact, this would be the same mechanism that makes a skunk smell.
It ends up that possibly BOTH Hypothesis #1 (study link) and 2 (study link) are correct depending on the individual who is peeing and the individual doing the smelling. In other words, there are some individuals whose urine does smell after eating asparagus AS WELL AS some individuals who may or may not be able to smell this foul odor.
The exact culprit compound is still unclear however. The top contestants are methyl mercaptan, thioesters, and asparagusic acid.
Regardless, there's no harm in eating asparagus at all whether you stink or not... or whether you can smell or not.
Read more about this phenomenon in the Boston Globe here.