A short sale is where the homeowner is asking the bank to accept a smaller amount than what they owe on their current mortgage. Buying a short sale is similar to buying a home directly from the seller. The only real difference is that when the seller accepts your offer, you still need to wait for the seller's current lender to also accept your offer.
Bank-owned/Foreclosure is owned by the bank. With a bank-owned property, the bank has already foreclosed on the property and the (previous) homeowner is no longer in the picture.
You can finance or pay cash for both short sales and bank owned properties, you will need to either need to have a pre-approval letter from your lender or show proof of funds with bank statements or investment statements.
The wait time can vary on getting a response on both short sales and bank owned, short sales are typically a slower process unless it is a pre-approved short sale. If you need to close quickly, I would not recommend pursuing a short sale. Bank owned can be a faster response but again the response times do vary depending on the bank.