How do I achieve those additional patterns seen in some of the demo images?

They are achieved through clever combination of several Honeycomb tiles textures with varying parameters.

Some use different tile sizes, some use different angles, some use different texture space UV offsets, and most are combined with different color parameters through a Color Mix node with varying Blending Modes to create masks.

Combine your own in new inventive ways to discover new different patterns that suit your particular artistic needs.

Here are some examples and the corresponding node setups to help you get started

What are those additional groups?

What the Hexagon XYZ node group?

Should I append that additional group too?

Can I remove it from my file?

Those groups are additional internal components of the main texture, they provide a way for it to work and eased the creation of the texture by recycling reusable modular components.

You don't need to worry much about them, they are appended/linked automatically with the main Cycles Honeycomb Tiles texture and should stick around as long as it is in use in your file.

Can I delete them?

No, you can not, as long as they are in use you should be unable to delete them. Blender prevents removing used datablocks by default, and if you did the hexagonal tiles texture would stop working properly anyway.

If you no longer use the Cycles Honeycomb Tiles texture in your file they should be discarded automatically.

Just erase all uses of the texture, save and reload Blender and they should be gone.

Can use them elsewhere/change them?

Sure, they can be recycled elsewhere if you know how to, it could yield interesting results.

You can append them independently on new files too. If you modify them in the same file you have the hexagonal tiles texture in use make sure you make a copy of the groups and rename them prior to modifying, so you don't risk breaking the inner workings of the original texture.

I still don't like them in my file, can I get rid of them?

If you don't want them around or find that the additional group names are polluting your file and/or your Add > Group menu you can manually enter the Cycles Honeycomb Tiles group and one by one ungroup every instance of those Nodegroups, so they are no longer used but the texture still works.

Make sure you start from the most nested occurrence to the least nested, to save repetition. Just be careful not to break it.