If you have a clear understanding of "the Nature of the firm" and "the Theory of the firm", you are pried to underline their similarities and differences.
Ronald Coase (1937) named his famous and seminal paper "The Nature of the Firm", which is, to my knowledge, the first ever on the issue. However, since decades, it has been more cited as a paper dealing with the Theory of the firm.
However, this only fact does not tell much about the differences and similarities between the Theory of the firm and the Nature of the firm.