American Heritage (AH) is one of the major American dictionaries, the other being of course MW (there's also Webster's New Collegiate, but it's very similar to MW). It's quite big because it has a lot of pictures, it's kinda like an encyclopedia.
The main difference between AH and MW (Collegiate) is that the first is prescriptive and the second (very) descriptive. This applies to :
1.Pronunciations, where AH largely ignores secondary/dialectal pronunciations, whereas Merriam Webster just goes on and includes everything (Think of "nuclear" pronounced "nucular," you can find that variant only in Merriam Webster)
Example, take a look at the pronunciation of the word "didn't" in
AH and then
MW. Also the word "particularly"
AH,
MW.
Note: MW's Third New International goes even further, and includes even more dialectal pronunciations, as in the word
"bag"
/'bag, -aa(ə)g, -aig/ where the second pronunciation can be found in the Inland North Dialect. I think that the Collegiate dictionary does, however, mention that in the introduction.
2.Definitions, where AH has a panel of language "experts" and gives you what percentage of them considers a particular usage acceptable, whereas M-W is much more lenient, with things like "some critics object...".
Example "fortuitous",
AH and
MW.
Besides this, AH's definitions are more "average joe"-oriented, whereas M-W's try to be very exact, and therefore can end up being more complicated (maybe that's why MW has been used in court). An example is the word "coy" in
AH and
MW
To try out more words on your own:
American Heritage and Webster's New Collegiate
Merriam Webster