Other threads have discussed the finances of the tsar and his relatives, but it's a very confusing picture to assemble, especially when trying to make comparisons to the wealth of the nobility.
The line between the tsar's assets and state assets was murky. He held much property in his own right, from which he derived private income. The official residences, though, were built and maintained partly with state resources. For instance, the army provided much of the labor force for Peter the Great's building projects.
I once read a report (and I cannot remember where) that shortly before WWI, the tsar's annual expenditures were around 25 million rubles while his income was only about half that. The difference was funded by the issuance of state bonds, sold largely to investors outside of Russia. So it appears that Nicholas kept his household of some 17,000 people going by a combination of personal income and a draw on state resources.
The definition of "household", though, is not a conventional one. It included the staffs of the Imperial ballet and theater companies, the curatorships of state art collections, and many other things that were controlled by the tsar but accessible to much of the citizenry.
The wealth of other members of the imperial family derived from a combination of personal property (which passed by bequest) and payments out of state coffers from a "civil list", such as was the practice throughout Europe and is still the practice in Great Britain. The amount of payments was determined in part by rank and in part by the tsar's discretion.
There was also a consideration of gender in the right to access assets. In her biography, Grand Duchess Marie Pavlovna the Younger mentions that she had to apply to her brother Dmitri for money to cover expenses, although she received payments from the civil list and held considerable property in her own right.
Even the wealth of the nobility, which was more clearly private, neverthelss often contained components deriving from state power. The tsars, as did their counterparts elsewhere in Europe, often conferred state monopolies on their favorites or on people who had provided extraordinary service to the crown or nation. The Yusopovs, for instance, derived much of their vast wealth from state mining concessions.
In one sense, the richest Grand Duke was the one who was most in the tsar's favor -- for in Russia, one way or another, everything came down to the tsar, and no amount of "private" wealth could offer inviolate protection or control of one's own affairs.
As Grand Dukes Paul, Michael, and others learned over the years, if you displeased the tsar, he could confiscate your property, exile you from Russia, confine you to an estate, revoke your right to raise your children.