Say the price of a stock is $3.00 a share. At 10:00am Bob places a buy stop order for $3.02. At 10:01am, Mary places a market order for the exact same number of shares. Right after Mary places her order and before it can be filled, the price jumps to $3.07 and continues to climb. Since it's above $3.02, Bob's order will be converted to a market order. But, which order would get filled first (i.e. for the cheaper price on the climbing stock price)?
Mary's market order places a priority on execution, not price so it is immediately routed to the exchange to be filled immediately.
Bob's stop order places a priority on price and it will be executed only if a certain price is reached. When that price is reached, the stop order becomes a market order.
If there are limited shares available at a lower price, Mary will have first dibs on them. Bob will also be filled at that price unless share price rises before his order reaches the head of the queue.