I understand the math behind the VIX calculation is relatively complex but I believe the concept is relatively simple. If I understand it correctly, the VIX is calculated in part by aggregating the prices of SPX calls and puts between 23 and 37 days into the future. To get the result (the VIX itself), is that aggregated price then compared to the current price of the SPX and the greater the difference (plus or minus, doesn't matter), the higher the VIX?
No, it's not that simple. The VIX is intended to be a measure of implied volatility in the market. Yes, the way you measure implied volatility is by comparing option prices (all else being equal, the higher the volatility, the higher the price of an option), so there is probably very high correlation between the VIX, and the price of options relative to the SPX (e.g. dividing), but it's not as simple as taking the total and subtracting off the SPX. The arithmetic difference is not as important as the relative difference.
Option prices encode expected volatility via a probability distribution of future SPX prices. The VIX is essentially a normalized standard deviation of that inferred distribution. Option prices don't give a single prediction for the future SPX price that could be "plus or minus, doesn't matter".
The closest thing to a single prediction is the expectation value of the distribution, which due to arbitrage is always tied to the current SPX price (adjusted for interest and dividends, similar to futures trading).