Actually, nobody prevents you from using tables for designing.
But common argument is: tables were introduced to provide structure to data. By "providing structure to data", I mean presenting some tabular data, like Excel sheet, for example.
Short Answer: current HTML5 standard, it's support across major browsers - allows mastering layout of different complexity without use of tables. Divs - are bricks of the layout - thus it's most reasonable to use them.
Long Historic Answer
Long time ago, in HTML 3.2 spec
] it was mentioned, that [tables] ... can be "used to markup tabular material or for layout purposes". But again, it was recommendation "The wisdom of past experience encourages us to separate the structural information in documents from rendering information. Mixing them together ends up causing increased cost of ownership for maintaining documents, and reduced portability between applications and media"