Was there any difference between a mammoth and a
mastodon?

What can you say about hairy elephants? Appearances
notwithstanding, they were shunted onto one of
evolution's sidings, rather than its trunk line. They
didn't survive to have peanuts thrown at them at the
zoo.

To most people, all hairy elephants look alike. They
can't tell a mammoth from a mastodon. But they were two
different animals and the distinction could be
important should you ever trip over a large chunk of
ice containing one of these extinct mammals. Mastodons
go back almost 40 million years and died out a million
years ago. They stood ten feet high and had straight
tusks. The mammoth, which emerged a mere million years
ago, had curved tusks, an extra outer layer of hair,
could reach a height of 15 feet and survived until
10,000 years ago.

Neither spoke English, used a cell phone or lived long
enough to symbolize the Republican Party.

I believe one can still buy a "piece" of one up in Hyder, Alaska! Of course, I was in the process of being "Hyderized" when I noted this fact...so maybe I'm a bit fuzzy on the details!

Did the Mastodon live much further back in time than the Mammoth?

Evidence shows that it 'mast-o-don'