In short, no. The Restoration nobility — which included both ancien régime nobles and those ennobled by Napoleon — had no formal legal privileges. The long answer, of course, is a little more complicated.
When Louis XVIII, younger brother of the guillotined Louis XVI, was restored to his throne in 1814, he issued a constitution for France, the Charter of 1814, which set up a constitutional monarchy. You can read the Charter online in English and French. The first section of the Charter, the first 12 articles, established a range of rights for the people and generally speaking enshrined some of the Revolutionary changes in law. Most specifically for your question, the very first article of the Charter read:
The next few articles elaborated:
They contribute without distinction, in proportion to their fortunes, towards the expenses of the state.
They are all equally admissible to civil and military employments.
The Bourbons were back, but the ancien régime was manifestly not. Though Louis XVIII might have preferred otherwise, as Louis wrote to his cousin, the King of Naples: "he had given a constitution to his peoples, because it was the spirit of the age, and it was always better to give one than to have one imposed on you."
Not that there wasn't some imposition going on: the great powers of Europe that had defeated Napoleon and restored Louis to his throne by force of arms insisted that France have a constitution before they would evacuate their armies from French soil. They wanted to prevent revolution and chaos from rising up again in France, and thought a constitution guaranteeing some of the Revolutionary changes was an excellent way to forestall further revolutions.
Beyond constitutional matters, Louis explicitly did not restore old noble privileges, despite wishes from some of the more extreme royalists to do so. Before the Revolution, nobles had famously been exempt from direct taxation; that exemption, which helped kick-start the entire affair, was not restored. (See Article 2 of the Charter, above.) Before 1789, one scholar estimated that around "5 percent at most of noble wealth was taken by state taxes... thereafter the uniform land tax was levied at approximately 16 percent." Nor did nobles regain their right to be executed by a headsman instead of hanging — all those convicted of capital crimes would continue to head to the guillotine, no matter how blue their blood.
In another example, peasants had previously faced a whole set of "feudal dues," which varied by region but included things like the requirement to perform unpaid road construction work a certain number of days per year, an obligation to turn over a certain share of their harvest to their lord, or a requirement to use their lord's windmill to grind their grain (and pay him for the privilege). All these remained gone, though peasants would remain extremely watchful for any perceived attempts to restore them.
Now, the complication to all this is that while noble birth no longer gave any official privileges in the new France, wealth most certainly did, and many of France's wealthiest men remained nobles. This included the usual ways in which wealth gives influence: nobles may not have been entitled to feudal dues from peasants working their lands, but now they could charge rent. Whether this made up for the lost feudal dues varied, though the answer seems to have been "no" more often than not; we have many reports of nobles bemoaning how they have been financially ruined by the loss of these old ways to make money. (Though "ruined" in some cases means being merely rich instead of super-rich.)
There was also an explicit legal privilege from wealth: the ability to participate in Restoration France's very limited democracy. The Charter set up an elected Chamber of Deputies, which had power to pass laws. But to vote for the Chamber, you had to be a rich man who paid at least 300 francs per year in direct taxes. That was about the richest 1 percent of French men — as if in the United States today, you had to earn $420,000 per year in order to vote. The threshold was even higher to serve in the Chamber: deputies had to pay 1,000 francs per year in direct taxes.
Of course, while nobles were a big share of France's wealthy enfranchised political class, there were other wealthy men in France who cleared the threshold: merchants, bankers, proto-industrialists, non-noble landowners, the upper crust of professions like lawyers and doctors, etc. In 1815, for example, the first elected Chamber of the Restoration was about 45 percent ancien régime nobles, 12 percent Imperial nobility, and 43 percent non-nobles. To be fair, 1815 was an unusually royalist Chamber; the next year's elections returned 34 percent old-line nobles, 18 percent Imperial nobles, and 48 percent non-nobles. Depending on the political winds, France's share of old-line nobles elected to the Chamber would fluctuate from election to election between as low as 10 percent and as high as 51 percent, but most usually was in the range of 35 to 45 percent.
As you can see, the nobility retained a great deal of political influence as a class (though they were far from unified), even if it was never as much influence as many nobles thought they should wield. An 1820 law gave the richest 25 percent of voters a "double vote," letting them vote for regular deputies but also for a special group of deputies elected only by the super-rich, and in 1830, King Charles X proposed limiting the vote only to these richest 0.25 percent of voters (but the proposal helped spark a revolution that drummed him out of office). The Restoration nobility had to share their power and (largely informal) privilege with not only the parvenu Imperial nobility, but with a rising group of wealthy non-nobles. (The real losers here were the poorest rung of the nobility, those who had little more than their title, which now meant almost nothing in a country defined by wealth.)
Because of this situation, some scholars have dubbed the France of the years 1814-1848 to be the age of the "notables," a category (originally meaning simply those the state should take note of) that encompassed the old nobility, the new nobility, and the wealthy bourgeois, but excluded peasants, the working class, and most of the middle class. It was an age of formal legal equality, but pronounced social inequality, even if the requirements to be a notable were based on wealth rather than parentage.
SOURCES
If you're curious about this period, I'd definitely point you in the direction of The Siècle, the podcast I host about France 1814-1914. In particular you might find Episode 7: The Charter to be useful background, though a number of my episodes touch on material I summarized here.
For a general overview of class and social life in this period: McPhee, Peter. A Social History of France: 1789-1914. 2nd ed. Houndmills: Palgrave Macmillan, 2004.
My information on the social makeup of the Chamber of Deputies came from: Beck, Thomas D. French Legislators 1800-1834: A Study in Quantitative History. Berkeley: University of California Press, 1974.
The anecdotes about Louis XVIII were from: Mansel, Philip. Louis XVIII. Rev. ed. Phoenix Mill: Sutton, 1999.