The stock market is like an auction with bid/ask based on who is willing to buy/sell at what price at the time. Overnight news and other world markets can affect whether people want to buy or sell when markets open the next day. So prices can gap up or down between one close and the next open. Often it has nothing to do with that particular company if there is good or bad news about a similar company, or the markets in general.
The same way a stock price can be one price at some point during the day, and something else a second later. The stock price is always the last sale. On most stocks there are orders waiting to be filled at the open. The opening price will be the first trade price.
The only time the opening price will be the same as the previous close is when the volume is very low. If there are no orders filled at open, then the last price would be the previous close.
Afterhours trading has nothing to do with the opening price. Afterhours trading is a separate session from regular trading. The spreads are usually very wide because of speculators and bargain hunters. But sanity return at the open.
The difference in a stocks closing and opening price is due to after hours trading. Which means exactly how it sounds, many stocks are available to trade in after hour markets, leaving you with a quote from the previous days closing, and not necessarily the current price.