When German Raiders Did Use Captured Ships
In addition to the other answers, there were a few examples of German raiders sending captured ships back early in the war. These were auxiliary cruisers, fast, long ranged merchantmen fitted with enough hidden weapons to overpower lone merchant ships. If they encountered a warship they'd disguise themselves as a neutral merchant. Many began the war outside of Germany and raided until destroyed. Germany had ten at the start of the war and they sunk hundreds of thousands of tons of shipping in the early war. As the war progressed, there were fewer German-held or friendly ports they could provision at, the tightening blockade made breaking out more and more difficult for a slow merchant ship, and increased Allied patrols made it more difficult to operate at sea.
Captured ships were rarely returned just to supply Germany, but rather to offload POWs and send captured documents home. Each came at a cost: a prize crew. A raider only had so many crewmen it could send to guard and man the captured ship. Each prize they sent home meant less crew for raiding. Eventually they'd run out of crew and have to return, likely never to raid again. A raider was more valuable staying at sea, tying up the enemy's resources looking for them, and sinking the enemy's supplies.
In addition, capturing a vessel took time; time when the raider would be vulnerable. The crew would have to be subdued. The ship surveyed. The machinery worked out and possibly repaired. All this time the raider is sitting stopped next to the merchant that possibly got a distress call off. A auxiliary cruiser's greatest strength is looking like an innocent merchant ship. If a warship or aircraft comes along and sees two ships stopped in the ocean where raider activity has been reported there is no question what is happening. It's safer to smash and grab: disable the ship, grab what you can, sink it, and exit the scene.
Instead, the supplies were used by the raider themselves, fuel and food being the most precious. Sometimes a raider would capture a merchant ship and use it as a floating supply dump until it was empty, then they'd scuttle it. Sometimes they'd supply other raiders and submarines as in the case of the tanker Ketty Brøvig which supplied not only Atlantis who captured her, but also the heavy cruiser Admiral Scheer, and the Italian submarine Perla.
Pinguin and Storstad
One of the most creative uses of a captured vessel was the Norwegian tanker Storstad by Pinguin, the most successful German raider of WWII. Pinguin captured the Storstad and found a nice remote spot to convert her to a mine layer (one of the astonishing things you learn in these histories is just how much work you can do on a ship while at sea) and commission her into the German Navy as the Passat. At the point in the war the Germans were still following the rules of warfare at sea and to use Storstad as a mine layer it had to be commissioned as a warship lest they be considered pirates. Together they quite effectively laid mines off South Australia, the two ships being able to cover more ocean.
With that job done, they returned her to her original merchant ship form and decommissioned her. She was crewed by German prize crew and Norwegian volunteers. Now she was a nice, innocent "Norwegian" merchant to scout for Pinguin, which she did effectively. A merchant ship that happened to be radioing position reports of nearby vessels that a warship might pick up was fine.
After sinking 11 ships Pinguin found herself with 405 POWs aboard. They were loaded onto Storstad and after using her to refuel other German raiders they were sent on their way back to France.
Pinguin and the Norwegian Whaling Fleet
Pinguin had a grand time with the Norwegians, capturing a whaling fleet intact was her greatest coup. This whaling fleet was operating for the British and Pinguin picked up their radio chatter. She waited until the factory ship Ole Wegger was transferring oil to the tanker Solglimt and then slipped alongside and quietly captured both and most of her whalers. Pinguin then quietly captured another factory ship, Pelagos and her whalers.
Rather than sink them the Norwegians were told to continue their work. They were working for Germany now and would be paid. At this point in the war Norway was occupied by Germany, and whalers like to be paid, so they did.
Pinguin then ran north-west at high speed for five days brazenly broadcasting signals home. The British, as expected, picked this up and, as expected, began searching along this new course. Her ruse complete, Pinguin returned to the whaling fleet who had been diligently working meantime, rounded them up, and headed off in the opposite direction.
Lacking prize crews for all 15 ships, but whale oil being extremely valuable, the oil was transferred to two ships and they were sent to France. The rest rendezvoused with the tanker Nordmark, supply ship Alstertor, and the captured refrigerated ship Herzogin. One whaler was converted to a minelayer, and the rest were sent to France with prize crews from Nordmark.
Pinguin was resupplied with ammunition from Alstertor and food from Herzogin. The captured Herzogin had been supplying a good portion of the German navy with fresh meat and eggs, but was running out of things to burn to keep her refrigeration plant running. Her bridge, lifeboats, masts and decks had all been torn up and burnt. After supplying Pinguin she was scuttled.
Examining the cases where Atlantis, the second most successful German raider, sent captured vessels back or used them for extended periods, gives an idea of the reasons they would do so.