Post-modern portfolio theory (PMPT) differs from modern (Markowitz) portfolio theory (MPT) in both how risk is measured and in how returns are distributed. The two theories are closely related and PMPT is a generalisation of MPT: MPT is PMPT with normally distributed returns and variance as the measure of risk.
As PMPT is more general than MPT there is no single formula: there is more than one approach to measuring risk and many to modelling the distribution of returns.
Post-modern portfolio theory uses measures of downside risk, by measuring the risk that returns will fall below a minimum acceptable return (MAR). This depends on the investor. This matters because risk is not necessarily symmetrical: the shape of the part of the distribution above the expected value may be very different from the part below it.
Because the MAR is investor specific, this means that there are are an infinite number of efficient frontiers, one for each minimum acceptable return. This means it more accurately reflects the reality that not all investors have the same aims or appetite for risk.
PMPT is far more complex than Markowitz portfolio theory. That is one reason why MPT used a simple mean variance model: it is much easier to handle mathematically and any other approach was impractical before computers became cheap and fast.
In general, using PMPT will not lead to dramatically better portfolios than MPT, but will improve returns somewhat and PMPT can tailor portfolios to investor preferences.
Measuring downside risk is also intuitively closer to most people's understanding of risk: it is the probability of making a loss, not that of making a bigger profit than expected. The distinction is not that important if returns are symmetrical, but if we accept that real life returns are often not symmetrical (especially true for volatile investments held long term) then the difference matters a great deal.
Not using the normal distribution also means that the distributions used can reflect other properties of real life returns such as fat tails.
The methods of risk measurement used in post-modern portfolio theory can also be used anywhere else that risk measures are used. It is certainly natural to apply them to valuation and performance measures: for example the Sharpe Ratio can be modified to become what is called the Sortino Ratio.