Since some businesses let you use your credit card without a signature, why do SOME still require a signature?

Well, the question is going to open up a big bag of worms. Did you know that you can dispute any charge to your card within 30 days, and they wont pay or incure the charge on your card. However, if it is a signed slip, they just match it with your signature(s) on file from the application and prior charges. Now, if something is unsigned, that would depend on the how the charges were incurred. Home solicitations, ( over the phone ), they are suppose to go through a series of questions to make sure you are who you say you are, problem is, these companies are more than happy to just charge it, and get the sale. That opens a whole legal battle on their part if someone did steal a card. Because they didnt do their due process to make sure it was legit. Now, if the "shipping" address is the same as the "billing" address , then they will win.
Your question can not be answered as easily as you asked because each situation has diifferent rules they go by. Gas stations, all tape their "pump" transactions, and would be very easy to determine if it was you, or not.
The answer is, there is no black and white answer, each purchase has different rules they follow.

Currently signatures on charge drafts are required for charges over $25.00. The merchant is taking the chance on not requiring signatures on smaller transactions.

The idea is it speeds up processing credit cards and it makes lines move faster. It also saves paper and paperwork/record keeping.

The risk and exposure lies with the merchant, who, without a signature cannot prove the charge was made by the customer if there is a dispute. Usually the card issuer will meet the merchant half way and not charge back the whole transaction since the amount is so low.

Someplaces just want to have the proof so if there is a problem they have the signature of the person that used the card.

The only problem you really run into is if the Credit Card company decides to fight you on it, and you cannot prove that you cancelled the card. But, once you call and notify the Credit Card company that the card was stolen or otherwise compromised, you are legally no longer liable for any charges made on it afterwords. Call the Credit Card company and tell them that a charge was made on a cancelled card and that you are disputing that charge, then do it again in writing by mail. Also, call the store and tell them that they made a charge to a stolen card, you did not authorize a $156 charge and ask them to remove the charge. As far as I know, once the transaction is complete, unless there is a written agreement to the contrary, they cannot come back and charge your card again without your permission. If they sold you the necklace and accepted the original $56 charge as payment, then the transaction is over as far as the store is concerned. If you still have your receipt for the original purchase, you can prove your position and I don’t see that there is much the store can do.

They just want to speed up so they can get more transaction going so more $$$. But like other say they are also taking the risk.

first on the Internet they ask for number on the back of your credit card that is your signature. second mail in most of the time you have to sign the order form. Third I feel you should have to sign on everything that way it will be harder for people to still your credit cards

Unfortunately some companies are still wayyy behind the times when it comes to the equipment needed to do purchases without a signature. The stuff needed to do that can cost quite a bit of money to buy and get set up. Thats usually why you would still have to sign at some places.
Though another reason some places require a signature is that the credit card companies will still pay a charge to the company... even if the card was stolen...IF it has a signature.

Probably because certain stores probably dont have the updated types of debit/credit machines that other places do. Thats the only thing I can think of.