The numbers on the bottles all relate to the thickness of the oil at certain temperature thresholds. With 0 being the viscosity of water, higher numbers mean thicker/slower flowing oils. In short, a 5w-30 oil flows faster than a 10w-40 oil.
The "w" in the viscosity indicates the oil's viscosity in winter temperatures. Generally, a faster flowing oil in colder temperatures is desirable because this means more internals get oil at startup faster than a slower flowing oil, reducing wear and tear on your engine.
In terms of gear oil, you'll see ratings like GL-3 and GL-4. The higher the number, the more lubricity the gear oil has, but also the more additives. At first, the main difference between the two was the EP additives comprising of phosphorus and sulfur compounds which will corrode metals like copper and bronze. Needless to say, as much as a better lubricant is good for your trans, the EP additives will eat synchronizers. Nowadays, the main differences between GL-3, GL-4, and GL-5 are pressure additives. These pressure additives handle closer tolerances between gears, but the issue this can cause is the additives are more durable than the gears themselves, slowly chipping away at the metals. If you see a GL-4 bottle that's also rated for GL-3, this means it doesn't have a worrying amount of additives. If it's a GL-4 bottle without the GL-3 rating, don't use it in a transmission calling for GL-3.
ATF+4 and ATF+3, on the other hand, are significantly simpler. ATF+4 was developed to succeed ATF+3 applications and degrade much slower. Where a transmission with ATF+3 might experience some shudder at around the 30k mile mark, the same trans with ATF+4 shouldn't experience shudder until 80-100k miles (depending on driving conditions). In short, ATF+4 is a complete replacement for ATF+3.