Advantage 1: Mixed effects models are typically the correct model for your data
Unless you have done completely randomized assignment of your treatments and taken only one measurement on study subjects, there is a good chance you either need a mixed effects model or need to remediate the situation (Murtaugh). Identifying when a the appropriate structure of a random effects model can be tricky, especially with observational data and we fairly commonly encounter models that are mis-specificed while reviewing papers and also in the published literature. Remediation of the problem can often involve including nuisance parameters that are not your focal interest, such as including site or year as a factor in a model (series of papers in Oecologial/Oikos on specifying ANOVAs for multi site experiments?). Remediation can also involve taking the average of subsamples, randomly selecting a single observation (example from meta-analysis I read ... I think was in Biological Reviews ... on parasites - Scandenavia dude who is controversial or some other Europeans - randomly chose one effect from studies in their meta analysis when there were multiple effect sizes reported. ) In some cases remediation can simplify the structure or interpretation of the model (See also the note in Bolker's book chapter). Much of this paper serves as an argument for the advantages of mixed models that are relevant to remedial measures