To be honest, I think it's both to test it, and as a publicity stunt - for something that's already as hyped as Halo 3, a public beta that is open to everyone (that is, the invites are available to anyone) will get the hype train rolling even more. From all accounts, Halo 3 is finished, and just needs a bit of tweaking and polishing here and there to make it the best game of the year. In this case, with a game that is the third in a trilogy, with an online feature that is probably built on it's predecessor, there's not that big a need for the network load to be tested, nor for them to see the public reaction - that's already pretty much known.

There are very few games I can think of that have had a public beta like this. Most seem to be things that specific people take part in (for example, the Sony Games Advisors Panel, or GAP), rather than something open to the masses.

The current Beta of Warhawk is apparently the full game. And those who are on it (once again, it's an open beta which, to be honest, is probaby Sony's response to the public beta of Halo 3) can play the whole thing... except it's only a month long.