The biggest difference is that a document can have multiple elements with the same class, but not with the same ID. An Identifier must be specific to a certain element within a document (i.e. full HTML page).
When applying CSS styling, some experts have recommended using classes over IDs to avoid specificity wars. CSS Lint (http://csslint.net/) recommends this.
The reason is that stylesheets are read from top to bottom, but IDs take precedence over class. This means that:
#stockexchange-id {
color: blue;
}
.stockexchange-class {
color: red;
}
Would color the text blue, even though red comes later. Here's a good graphic explaining specificity: https://stuffandnonsense.co.uk/archives/images/specificitywars-05v2.jpg.
If everything has the same specificity, and most styles are applied via a single class, it makes the CSS easier to reason about, since it will read from top to bottom without extra styles being mixed in different orders. It also makes it easier to override classes, which can be useful.
Finally, if you style based on classes, you can re-use the styling on the same page by applying the class to another element on the page. With IDs, since they are unique, you cannot do this. Even if you think an element is unique on a page (e.g. a Buy Button), this may not always be the case. A designer may request the same element again later in the page.