The Germans did not, in fact, start WWII with more submarines than the Allies. The German Navy had a total of 57 submarines, most of which were the smaller Type II coastal boats. The RN had the same number, though twelve of these were obsolete WWI designs. The large French and small Polish submarine forces must also be added, resulting in an Allied submarine force that was more than twice as large as the German one. Despite this, the Allies did not make the submarine the centre of their naval war in the same way the German Navy did. The RN didn't build submarines to the same level as the Germans did, and so their submarine force would end up being smaller than the Germans over the course of the war.
The main reason for this was that they did not need to, unlike the Germans. The only way the Germans had to cut the supplies of fuel, food and strategic materials that Britain (and, to a lesser extent, France) depended on was through submarines. The Royal Navy could bring overwhelming force to bear on any surface raider that attempted to slip out into the Atlantic. This made surface raiding unacceptably risky. Submarines, meanwhile, could avoid the Allied patrols, and attack convoys with less threat from Allied hunting groups or convoy escorts. Losing a submarine hurt much less than losing a large surface raider, yet the submarine could be equally effective in aggregate. For the British, though, this was much less of a concern. The German Navy could not effectively combat the British surface blockade. Any ship that wanted to carry supplies into Germany would likely be stopped, searched and captured by a British ship. There was nothing the Germans could do about this. Other supplies came in overland, from Germany's allies and conquests, or until 1941, the Soviet Union. These could not be stopped by Allied submarines. Submarines, therefore, were much less useful to the Allies in the Atlantic. As such, they built a much smaller submarine force, redirecting the efforts of their shipyards into ships that were more useful to them like ASW escorts.
That said, the Allies still made effective use of submarines. The Royal Navy carried out constant submarine patrols in the North Sea. These aimed to hunt German raiders and blockade runners, and to cut the sea routes between Germany and Norway. The RN also carried out an effective submarine campaign in the Mediterranean, targeting the sea lanes between Italy and North Africa. The Soviet submarine force was the only portion of the Soviet fleet that could effectively act in the Baltic and Black Seas, due to the persistent air threat. Finally, the USN, RN and Dutch navies carried out a submarine campaign against Japanese trade in the Pacific Theatre. Here, the strategic situation favoured submarines. Japan was highly dependent on imports of strategic materials, especially oil. The Allies had, after the Japanese conquests of early 1942, no bases near the main sea routes used by Japanese trade. The Japanese navy had strong surface and air forces that could easily attack surface raiders. The Japanese ASW establishment was also much less effective, due to poor doctrine and a lack of WWI experience, than the British one.