What is better Modern day English or Old English of Shakespeare Why?

Technically, there is no "better" version of the language, because language is evolving every day. Modern English is what we speak, but Shakespeare did not actually speak (or write) Old English, he wrote in Early Modern or Elizabethan English. Old English (except for a few often-used Words that have survived) would have been unintelligible, as it is a form of early Germanic languages.

The grammatical differences between different dialects of the same language do not make one version better than another: "thou art" is no better or worse than "you are" and "you are" is no better or worse than "y'all are". Shakespeare's use of the language is better not because of his grammar but because of his extensive vocabulary, his use of figurative language, and his close attention to rhythm and the sounds of Words.