Both of these views are a form of Calvinism which rejects the Dortian view that the atonement is limited. Indeed, if one merely considers them by what they affirm and don't affirm from TULIP, then they would be considered the same thing, or at least one being a type of the other. However, I think that there is a difference between Amyraldism and what is currently referred to as five point Calvinism.
The distinction lies in the character of the views. Four point Calvinists are essentially people who have been convinced of Calvinism, and embrace the label passionately, but recognize that the doctrine of limited atonement is completely and utterly contradictory to Scripture. So therefore they reject it, and merely affirm the opposite.
Amyraldism though is based on the beliefs of Moses Amyraut, and who had attempted to fully integrate a general atonement theory in with Calvinist teaching. Amyraut felt that he was following the beliefs of Calvin himself, and there are many who believe that Calvin did in fact teach a general view of the atonement.
...it is at least interesting to note that John Calvin himself did not believe in this doctrine [Limited Atonement]. In 1979 researcher R. T. Kendall (b. 1935) published a powerful argument that Calvin did not believe in limited atonement: Calvin and English CAlvinism to 1649. -Roger Olson, Against Calvinist, (Grand Rapids, Michigan: Zondervan, 2011), 145.Because of this, we can see a big difference from the common Four-point Calvinist and the Amyraldian: one is a pragmatically formed view which is unconcerned with logical consistency, while the other is a carefully articulated theological view.
This brings us to another distinction. Amyraut integrated a general atonement by use of high decretal theology. I'll explain this fuller next week, but today it suffices to say that decretal theology involves technical theological languages which is not used by the average person. Therefore, for most Calvinist laymen, Amyraldism is rather inaccessible, or at least requires additional intentional study. On the other hand, anyone who understands what TULIP is can appreciate four-point Calvinism. You merely drop the 'L'!
In summary, while one may simply see Amyraldism a form of four-point Calvinism, I think four-point Calvinism can be considered a distinct phenomenon, where Amyraldism is a careful and technical articulated position, four-point Calvinism is a guileless attempt to be biblical while holding to Calvinist teaching. Where Amyraldism can be admired for its sophistication, four-point Calvinism can be admired for its honesty and naivete.
Next week, I'll be looking more exactly at what Amyraldism is, and after that we'll start talking about their true value in theology, and how they relate to Calvinism and Arminianism.