What's the difference between hyperreal and surreal numbers?

There are many non-isomorphic non-standard models of reals; any of them can be called hyperreals, although one specific model (the ultrafilter construction on $\mathbb{R}^\mathbb{N}$) is often called "the" hyperreals.

Models are generally taken to be sets. The surreal numbers are a proper class: they are "too big" to be considered a non-standard model of the reals in this sense.

But to some extent, we don't really have to insist on models being sets: with suitable set-theoretic axioms, I believe the surreal numbers are also a non-standard model of the reals. In fact, they would be the largest model.

If we pick one particular (set-sized) non-standard model -- e.g. "the" hyperreals -- then we cannot compare its elements to surreal numbers directly. First, we'd have to choose a way to embed the hyperreals into the surreals. There isn't a unique way to do this. In fact, there is a vast number of ways -- an entire proper class of embeddings! (I believe) we can choose to make any particular surreal a hyperreal number by choosing an appropriate embedding.

According to the recent work by Ehrlich, the surreals are also a subset of the hyperreals. More precisely, maximal class-size fields of the surreals and the hyperreals are isomorphic.

When one wishes to trim down one's model to set-size proportions, the advantage of the hyperreals over the surreals becomes obvious, because the hyperreals possess a transfer principle that makes them useful in analysis and other fields of mathematics.