You have asked a very interesting, complex, and controversial question. It is not a easy one to answer, nor are there definitive answers. I'll confine myself to the German and US boats, the ones I know best.

Comparing the US Gato fleet boats to the German VIIC compares the types that did the bulk of the fighting for the two navies. (Though the German type IXD2 is more comparable to the Gato in size - that doesn't change much the conclusions of my post).

First, let's compare the boats general surface characteristics. The Gato class was generally superior to the VIIC on the surface. Its greater size meant higher speed, greater range, better crew habitability, and a more stable gun platform.
The flexibility and robustness of the 4 engine indirect drive system of the US Gato's made engine failure caused aborts much rarer than in the direct drice, 2 engine U-boats. The longer range was a big advantage: even without milch cows or mid-ocean refueling (routine with U-boats) rarely did Gatos with torpedoes aboard and targets waiting head home because of a lack of fuel, a common problem with the VIIC: it was just too short ranged for the Battle of the Atlantic. On the other hand, surface gun actions by subs (apart from US plinking at fishing sampans - a widespread habit, especially in the later war years) were rare in WWII. And, despite air conditioning, and better crew quarters, the US crews had harder times completing their generally shorter patrols. They suffered more often from rotten food or spoiled water. The bigger US boats had bigger profiles, making night surface attacks harder (before radar made this difference moot). So give the surface characteristics to the US boats, but with reservations.

Under the water, the German boats take the lead. The smaller VIIC had greater maneuverability, quietness, and greater diving depth. But, though greater depth seems a good advantage, it was actually more important to be able to submerge quickly than deeply. Most U-boats were destroyed on the surface or just under it, not at great depth. But still, the VIIC could submerge quicker than a Gato, but not by much: a well trained crew could get a Gato under in 35 seconds, vs maybe 20 or so for a U-boat. 'Under' was 60 ft for a Gato, and 45ft for a VIIC, due to the different sizes of the boats. Give this edge to the VIIC.

As for weapons, the German torpedoes were clearly leaders. Though both sides struggled with torpedo problems, no one equalled the plethora of problems, and beauracratic inertia in solving them, of the Americans. US subs had torpedoes that broached, were duds, ran under targets, even circled, with desparing regularity. Their warheads were smaller. American electric torpedoes, a copy of the German version, lost something in the translation, and was full of problems. The US acoustic sub torpedo, the 'cutie', just plain didn't work, while the German T-5, well, sometimes worked. (Not to be confused with the US 'Fido' air dropped ASW acoustic torpedo, that proved VERY deadly against U-boats - US subs never used this weapon).

As for mechanical systems, again a clear German lead. German sonar, optics (periscopes), machinery, from hull construction to engine design, was ahead of the Americans. The Germans even had schnorkels, though in reality the schnorkel was not very useful. A sub using a schnorkel could only make 4 or 5 knots and was blind (periscope and sonar unuseable).

There is only one last category, but in some ways, the most important: electronics. Here the US boats enjoyed a huge, decisive, and commanding difference. US boats were equipped with a variety of radars, radar detectors, jammers, and communication devices that U-boaters could only dream of. At a time when U-boat commanders were struggling with makeshift detectors (that didn't work) made from wood, string, and wire, US boats sported a veritable forest of antennas off of their periscope shears. While U-boats were always at the electronic disadvantage to allied ASW, the opposite was true of US Gato boats fighting the Japanese. When many allied ASW ships had radar, Doenitz still claimed radar was impossible to mount on ships (let alone aircraft). When Allied ships were using direction finding to home in on U-boat radios, Doenitz claimed this was impossible. But most of all, the US boats had highly effective radar (the SJ) and very effective radar detectors. This offset all the disadvantages above, including slow diving times (radar gives better warning times), bad optics (use radar in attacks), etc. Towards the end, US boats even got the ST periscope mounted radar - one radar 'ping' and you have the exact range to the target - far offsetting any disadvantage in night attack scope design, and taking the guesswork out of attack solutions. And, of course, the US boat's radio codes were not compromised, while the U-boat 'Ultra' was read with increasing ease as the war dragged on. Many, (most?) U-boats were defeated electronically, by enigma intercepts, direction finding on radio emissions, or radar detection. The allies' complete mastery of the electromagnetic spectrum over the U-boat is the major reason for their defeat, a huge factor in the US subs victory over Japan, and the major difference between the US boats and the U-boats in combat effectiveness.