Short Answer: You're both correct. Which date to pick for ending the French Revolution is a matter of opinion.
Your friend is not wrong. The downfall and execution of Maximilien de Robespierre is considered by many to be an end date for the French Revolution.
For many historians, the end of Robespierre coincided with the end of the Revolution itself.
- Huet, Marie Hélène. Mourning Glory: the Will of the French Revolution. University of Pennsylvania Press, 1997.
This viewpoint has had a lengthy pedigree. Robespierre's execution was where the great Jules Michelet concluded his foundational narrative of the French Revolution. Later prominent historians Arlbert Mathiez and Jean Jaurès, identified the Thermidorian Reaction as terminating the Revolution.
The classic interpretation, going back to Jaures and Mathiez, had presented the coup of Thermidor as the end of the Revolution, aborting both its social premise and the resolute elimination of those who threatened it ... When [Robespierre] fell [the Revolution] ended, in the sense at least that this Jacobin discourse was abandoned, along with the limitless ambitions which it articulated.
- Haydon, Colin, and William Doyle, eds. Robespierre. Cambridge University Press, 2006.
Having said that, you are not wrong either. Napoleon's coup in 1799 has also been identified as a end date for the Revolution. The main reason for picking this event is that it marked Napoleon's rise to power. He reversed many of the principles of the Revolution and eventually installed himself, de jure, as a monarch. Since the Revolution proclaimed equality and democracy, it could then be seen to have ended in a personal dictatorship under Napoleon. It therefore follows that his seizure of de facto power in the Coup of 18 Brumaire may be cited as the end of the Revolution itself.
Why is this different from the Directory's establishment? It isn't, not really. As the previous section says, the Thermidorian rise to power with the execution of Robespierre is also considered to have ended the Revolution. What is different with Napoleon, is that he managed to stay in power (with a brief interruption) until definitive general peace in Europe was achieved. By his overwhelming dominance of the period, Napoleon lend his name to both the period and its wars.
In this sense, Napoleon's rise to power thus marks a convenient turning point in the narratives of the period. But this is by no means the only possible date of either beginning or conclusion for both the French Revolutionary and Napoleonic Eras.
Ultimately, when to date the end of the French Revolution boils down to a matter of judgement. As the respected French Revolution historian François Furet explains:
For the same reasons that the Ancien Régime is thought to have an end but no beginning, the Revolution has a birth but no end ...
Even in the short term, it is not easy to 'date': depending on the significance the historian attributes to the main events, he may encapsulate the Revolution within the year 1789, seeing in it the year in which the essential features of the Revolution's final outcome were fixed, when the final page of the Ancien Régime was turned - or he may go up to 1794 and the execution of Robespierre, stressing the dictatorship of the Revolutionary committees and of the sections, the Jacobin saga and the egalitarian crusade of the Year II. Or he may use 18 Brumaire 1799 as the terminus, if he wants to the acknowledge the extent to which Termidorians had remained Jacobins, and include the government of the regicides and the war against the European monarchies. He may even integrate the Napoleonic adventure into the Revolution, perhaps to the end of the Consular period, or to Napoleon's Habsburg marriage, or even to the Hundred Days: a case can be made for any of these time frames.
One could also envisage a much longer history of the French Revolution, extended even farther downstream, and ending not before the late nineteenth or early twentieth century. For the entire history of nineteenth-century France can be seen as a struggle between Revolution and Restoration, passing through various episodes in 1815, 1830, 1848, 1851, 1870, the Commune and 16 May 1877. Only the victory of the republicans over the monarchists at the beginning of the Third Republic marked the definitive victory of the Revolution in the French countryside.
- Furet, François. Interpreting the French Revolution. Cambridge University Press, 1981.