Ori, Nori, and Dori
These chaps were of Durin's line, but presumably remote enough that they didn't figure into the (simplified) family tree. This is revealed in a footnote to the Dwarvish family tree printed in Appendix A:
Of the other companions of Thorin Oakenshield in the journey to Erebor Ori, Nori, and Dori were also of the House of Durin
Return of the King Appendix A "Annals of the Kings and Rulers" III "Durin's Folk"
Bifur, Bofur, and Bombur
This is unknown, but there was a kinship between them and Thorin, mentioned in the same footnote I referenced above:
[M]ore remote kinsmen of Thorin: Bifur, Bofur, and Bombur were descended from the Dwarves of Moria but were not of Durin's line.
Return of the King Appendix A "Annals of the Kings and Rulers" III "Durin's Folk"
They most likely accompanied Thorin because of that Kinship. However, it's worth noting that this isn't the first time the other clans have assembled for the benefit of the Line of Durin; in Appendix A, it's said that they came to help recover Khazad-dûm from the Orcs:
Durin's Folk fathered all their host, and they were joined by great forces sent from the Houses of other Fathers; for this dishonour to the heir of the Eldest of their race filled them with wrath.
Return of the King Appendix A "Annals of the Kings and Rulers" III "Durin's Folk"
Is there a difference between the House of Durin and other Dwarves?
Not a significant one it's principally historical. All Dwarves are descended from one of the Seven Fathers, which is where the clan distinction came from.