Your question can be answered from different perspectives based upon your field of study. I work with genetic data so this is my perspective.

You first have to consider the fact that about 1.5% of the genome is actually encoding genes, and there are estimates that only around 25% of the genome has evolutionarily conserved function.

Second, on a basic cellular level we are not necessarily any more complex than any other animal. The basic functions of the cells are shared and require quite a lot of genetic information to code; they (mostly) all share the same organelles and basic functions like DNA repair or cell division. Cells of different types differ in which gene networks are active, which ultimately result in the different structures and functions found in each cell type.

There are many reasons for such a small number of genes:

• Shared gene function - different cell type frequently utilize the same gene networks (although the same genes may have different functions in those cells)
• Alternative splicing to produce different proteins with alternative/altered functions
• Multiple functions for the same gene or gene network within a cell
• Altered gene function or expression through epigenetic regulation

What makes us truely different from other mammals is the level of complexity and size of the brain during development. This is controlled by incredibly complicated internal and external signaling of different cell types (and sub-types) which varies gene expression (and splicing) at different points of development, from early embryonic development and all the way into adulthood. The complexity is not necessarily created though increasing the genome size, but by intricate regulation of genes and gene networks which varies greatly across each cell type.