From what I understand both these documents contain formulas to calculate arc flash. There is a vast difference in price and if they both have the same formulas why get the more expensive document, IEEE 1584?
What is the difference if any between the formulas?

NFPA 70E doesn't give you formulas. It gives you charts and encourages you to do a study using someone else's formulas (such as IEEE 1584).

Polar is referring to annex D of the NFPA 70E

Polar, please review the informative Annex D section D.1 - Table D.1 for a brief explanation of the limitations and parameters of the different sets of equations (different methods) that are commonly accepted as means of calculations of the thermal energy released during an Arc Flash event - or cal/cm^2. You will find further information in each of the sections dedicated to each calculation method. You will find that the commonly called "70E equations" - (actually the section D.5 Doughty/Neal/Floyd or "DuPont" equations) are useful only for equipment operating a less than 600 VAC 3 phase with limited bolted fault currents ( i.e transformers of about 500 kVA to 2500 kVA). The IEEE 1584 equations take into account other variables such as "arcing current" instead of bolted fault currrent, bus gap spacing (both phase to phase and Phase to ground), and others , including utilization of much greater fault current and higher voltage levels, that have been found ( by some actual arc flash testing - ie starting arcs in a controlled configuration and determining energy release) to affect the thermal energy released in an arc event. In essence, you pay for greater accuracy in estimation of the thermal energy release.