If not what was the reasoning for not using/destroying them?

Well their are several.

One is that using enemy weapons tend to draw friendly fire as their guns and cannons sound very different than yours. For instance a German MG42 sounds very distinct compared to a Soviet DPM and if you heard one firing from a dugout you would probably would not think twice about tossing a grenade in there rather than pop your head in and see who is firing.

The same can be said for tanks and artillery. See when they capture them they mark them like this but even then, if you are a gunner with only a split second to react try seeing that cross from a half kilometer away, at evening light, half obscured with smoke, not easy. Same goes for bombing from the air or from spotters calling in artillery, soldier are trained to recognize profiles so you couldn't rely on a coat of pain to protect your tank or truck or cannon or plane.

Finally their is training, maintenance and ammunition. When equipping a unit with weapons you have to think long term. What happens when a spring in your gun snaps or a pipe in your tank bursts? Many weapons have dozens of unique parts that they need to function, whilst your own gear is supported by the factories who churn out spare parts captured gear is not and even minor parts breakage becomes a massive problem. Ammunition is another factor, sure you can capture enemy stock piles but in battle ammunition depletes very fast and without continuous supply even a crack fighting force can become useless. Finally if your soldiers are not trained on the weapons then they'll not use them effectively or they'll become useless at the first sign of trouble like a weapons jam. It is often better to stick to using even slightly inferior gear than replacing it with something that has the above issues.

If so

Definitely so. Captured weapons saw a lot of mileage during the second world war in many hands and this for many reasons.

Take partisans and resistance fighters, they often used and supplied themselves with enemy weapons almost exclusively. Being trapped behind enemy lines or in occupied territory you take what you can get. Also take anti partisan or occupation troops as well used a lot captured gear. The Italians, French, Yugoslavians, Polish ect were often times fighting against weapons that once equipped their armies. Since fighting resistance groups is low intensity work that uses up less ammunition and puts less strain on equipment than front line fighting it is the logical place to use it. Often times occupying troops got captured weapons which freed up domestic gear for the guys serving on the front line.

Another big example is the Germans who conquered whole entire countries. They're army was always overstretched and in short supply and captured goods eased their industrial burden. French trucks, tanks, rifles, machine guns and the like played a major role in all aspects of the war. They used French trucks and tractors to move supplies in their invasion of the Soviet Union. Early war French tanks were used effectively in the early stages of the Soviet operation and when they were deemed obsolete their chassis were often converted to self propelled gun platforms or supply tractors. A couple thousand pieces of French artillery was re-bored for German ammunition and used as anti tank guns as well. Even at the end of the war foreign guns were used to equip Germany's desperate home militias. The Germans did similar with the Italian gear they took when Italy switched sides. I made a long post about Germany and captured equipment here.

The Finnish though, they perhaps take the cake. They used many of the same weapons and munitions as the Soviets and were masterful scroungers. They for instance had a shortage of domestic light machine guns but captured so many Soviets DP guns they actually went ahead and adopted it as their main LMG. They also used the same types of Maxim Guns's and Mosin Rifles as the Soviets so it was easy to simply add captured stockpiles into their supply lines. Finland had little manufacturing capability, their army practically ran on Soviet gear. This site has lots of in-depth information on Finnish weapons.

Using captured weapons was always a tight rope walk but their was some tricks to it. Segregating captured weapons, IE: equipping say an entire Battalion or Division with captured weapons and making sure they don't cross paths with friendly units to lessen friendly fire. Producing enemy ammunition, like with the Soviet PPSh-41 submachine gun the Germans actually went ahead and produced 7.63x25 Mauser which fit the gun, another method is re-chambering a gun to accept your bullet, or adding your own cannon to a tank. Keeping weapons working can be accomplished through cannibalization which is basically turning lost vehicles into spare part kits. Say a gear breaks on a tank, well you retire that tank and if something else breaks on another tank like a tread then you take it off the retired tank. It is a crude solution but it works for a while.

I thought I had answered a similar question not too long ago, took a bit of digging, but here it is.

The Soviets were allowed to use captured tanks until they broke down, but even then the practice was far less common than it ever was for, say, the Axis. The use of captured small-arms on all sides must be considered negligible and anomalous, with a few exceptions - such as the German's mass-usage of PPSH-41s due to captured store, and the re-equipping of 3rd-rate Ost units with captured weaponry. The Allies, and the Americans and British in particular, never had to resort to this; even 2nd line units like the British Home Guard could often get top or near top quality equipment; such as being armed with older 1917 pattern Enfields.

For captured German material, there is one exception that proves the rule that the Allies generally had no need to impress German weaponry: the Panzerfaust. This antitank weapon filled in a niche that was relatively absent in Allied stocks. Although this photo is of poor quality, American troops found the Panzerfaust to be superior to the Bazooka. The Panzerfaust was comparatively easy to use and was portable. Furthermore, unlike routinely using a weapon like the MG42, the Panzerfaust's single use lessened (although did not eliminate) friendly-fire incidents as there was only one sonic fingerprint. Moreover, the one-shot nature of the Panzerfaust meant that one of the main rationales against using captured equipment- spare parts- was less of an issue as it came with its own ammunition. Their relatively small size made them ideal for emergency weapons for front-line units and they could either be used for bunker-busting or anti-tank duties. The virtues of the Panzerfaust- ruggedness, simplicity, mass-production- were generally not the same qualities that characterized other pieces of German equipment, as the other commentators have pointed out in this thread. The lack of a real Allied equivalent man-portable antitank weapon meant that there was a demand on this particular piece of equipment on the frontlines.

Edit added a little on the one-shot nature of the Panzerfaust

Source

Rottman, Gordon L., Johnny Shumate, and Alan Gililiand. Panzerfaust and Panzerschreck. Oxford: Osprey Publishing Ltd, 2014.