Why do complex organisms need specialized cells?

Because these complex multicellular organisms have many functions and are interdependent with other cells to survive. Also, because of mitosis the organism is able to grow and communicate with other surrounding cells forming a bigger and organized system of cells. 

A Eukaryotic cell contains a nucleus; more than one chromosome and is typically a multi-celled organism.  Both plant and animal cells are eukaryotic cells.
The advantage is that eukaryotic cells have a nucleus which functions formally as to store the DNA of the cell which can translate into many functions such as cell division and work structure between cells that forms the hierarchial function to tissues.