Nope, they're different. There have been three Graveeminds so far - the Timeless One's Flood-infected form in Primordium, the Great Gravemind in Silentium, and the Gravemind in Halo 2/3. While these beings are the same in many ways through their shared memories within the Flood itself, there has been plenty of evidence to show us that there is a degree of individuality in Graveminds - particularly in how the Gravemind in Silentium is singled out as the greatest of all its kind for having consumed 10,000 worlds and brought galaxies to an end. The Gravemind in Halo 2/3 possesses the memories of the Great Gravemind and the Timeless One before it, but is not necessarily the same.

Great Gravemind?

Just what I like to call it, since Forthencho singles it out as "the greatest of them" for bringing down galaxies and consuming over ten thousand worlds. Just an easier way to distinguish it when talking about Gravy on more of an individual basis.