OrthoMCL was introduced in 2003 keeping in view of several things that COG did not have then. The rationale, however is the same and widely applicable for both excepting that yeast records were not grouped in COG.
Second, COG is specifically constructed based on BLAST reciprocal blast hits making phylogentic relationships, further bringing false positives and extensive gene duplication records seen in COG.
At the end of the day, what matters us is to predict the domains that harbor complete eukaryotic genome further reducing the functional redundancy whilst associating the clusters.
Ortho-MCL uses an all-against-all BLASTP followed by a Markov Cluster algorithm to cluster putative orthologous protein sequences ( within a species or between several species). You can also use Tribe-MCL which is similar but easier to use.
COG and OrthoMCL are both based on blast result. COG merge adjacent triangles of Best hits blast. OrthoMCL use MCL clusters. For more information on orthologs you can read for example :
Altenhoff AM, Dessimoz C (2012) Inferring orthology and paralogy. In: Anisimova M, editor. Methods in Molecular Biology: Evolutionary Genomics Vol. 1. Clifton NJ: Springer, Vol. 855. pp. 259–279. doi:10.1007/978-1-61779-582-4_9.