A jug is a vessel for liquid that is made out of clay,ceramic,glass, or metal. They Have a handle and a curvy body, The bottom being round and the top tapering to a spout.

A pitcher is a glass or plastic vessel for liquids that is shaped like a cylinder and has small spout and a handle.

A jug can also refer to the big water pots they carry on their heads in africa/egypt. Also we call the container milk comes in a "Milk Jug" while it is never a "Milk pitcher". And yes, please aviod saying "Big Jugs" as that is sexual slang for breasts. We use both in the USA and they are different objects,I know because i have worked for the food indistry for 3 years.

And just to lower the tone slightly.... jug tends to be used more in the singular. 'Jugs' is an informal word for breasts, so you can't make a comment about 'large jugs' or 'nice jugs' without having somebody snigger. Or snicker. Depending on the variety of English you use. Sorry, Doreen, another difference there.

It depends on whether you the speaker/writer is using American or British English.

In American English, a container with a handle and pouring lip which you would bring to the table to serve drinks from is known as a 'pitcher'. In British English, this is known as a 'jug.'

In American English, a 'jug' is a large, sealable container used for storage.

I do not know the official difference, but to me, a pitcher is a more specific type of bottle that you use for pouring drinks. A jug could have more uses. Again, that is just how I think about it. 

Tangentially, do people know the (U.S.?) English saying,

"little pitchers have big ears?"

The handle of a pitcher is called the "ear." The phrase is a warning: "Be careful what you are saying, there are children in the room." The idea is to warn people that they are starting to stay things that children shouldn't overhear, without the children themselves understanding the warning.