What is the largest and heaviest payload ever flown?
Source: Popular Mechanics

The Antonov-225 currently holds a number of records...


Heaviest maximum take-off weight at 710 tons (1.4 million pounds)
Total air lifted payload of 558,590 pounds
Single item payload of 418,830 pounds
Longest wingspan of 290 feet for any aircraft in service (Stratolaunch and H-4 Hercules had larger wingspans, but neither are currently in service)


The A380 by comparison has a maximum take-off weight of around 1.2 million pounds (600 tons) and is 260 feet wide. 

For helicopters, the heaviest payload holder currently goes to the Mil Mi-26


Source: Wikimedia

It has the world record for a payload lifted to 2000 meters (about 6500 feet) of 125,153 pounds (about 62 tons). 



As for the "largest" I'm not sure you can separate that. The Antonov 225 holds the largest single item payload referenced above, so that is held by weight. If you are asking about physical size, that could mean a number of things. Longest, widest, tallest, etc. 

The Airbus Beluga held the world record for the most "voluminous" cargo ever lifted when it transported a chemical tank for a marine vessel.