If I have current earnings for the SP500 of $160, should trailing or forward P/E be used to calculate the SP500's price?
What is the difference in using one over the other?
The answer depends on what the future price for the Index will be used for.
Trailing Earnings, or Trailing P/E is used as a consistent and quite reliable metric for Index performance relative to its long-term historical trend, and as such, it is the measure to use when a more quantitative metric is required.
Forward Earnings or Forward P/E is a forecast, an educated and informed estimation. As such, it is useful when attempting to asses Risk. However, Forward P/E depends on factors that are generally not under anyone's control, such as future interest rates, future commodity prices, regulatory norms, geo-political forces, etc. As Master Yoda would say: "The future, easy to see it is; but the future is always changing."
For the purposes of investment strategy, both must be used. Historical P/E will be used to ascertain where current Index price is relative to its historical trend, and Forward P/E will be used to ascertain Risk -- at the same time.
There is no "answer".
Some traders happen to like to look at the forward P/E.
Some traders happen to like to look at the trailing P/E.
Many traders have utterly no interest in the P/E, and have never looked at it for any reason.
Most basically, your sentence says something like "can I calculate the price using..."
You can't, in any way, "calculate" the price of a stock. It's just what someone happens to pay last in a trade.