Random Mutagenesis vs Directed Evolution as Strategies to boost expression
Well, you already mentioned it. Random mutagenesis (with UV or chemical mutagens) targets the entire genome and you select the organism for certain traits.

In a directed evolution experiment, you are just targeting a gene or a group of genes. With the progress of synthetic biology, it might be possible to synthesize small genomes (already shown by Craig Venter and group) and apply directed evolution methodology genome wide. This would still not be considered directed, as the mutations are actually random over the genome. 

The pros and cons depend on what exactly you aim to do.