Before you read this, I will say that I have a very low opinion on Maximilien Robespierre, so this post might be very partial.
I'm not sure that Robespierre would have "cleansed" France of all its monarchists. A great number of them are probably going to go into exile for safety reasons but I could see "underground" monarchist societies try to topple the Republic.
I'm sure of one things though: if Robespierre remains in power, a lot of heads are going to roll. The man brought the Terror and I'm not sure he intended to abolish it for as long as France was at war with itself (The Vendée was still in uproar, the Bretons were highly anti-revolution and I think there were still a few federalists rebellion) and with Europe (First and Second Coalition War). This means the Guillotine will probably be used more than OTL.
The main powers of the Coalition would be as okay with Robespierre leading the French Republic as they were with Napoleon OTL ("A Robespierre on horse" for some). As a matter of fact, I think Robespierre would be even less liked by the Europeans than Napoleon as he would look more dangerous. Napoleon did kill thousands people but it was out of warfare and it's "excusable" but Robespierre killed thousands of people because he made laws that resulted in a lot of people being sent to the Guillotine. In the European's eyes, a longer-staying Robespierre will probably appear more as a butcher than Napoleon OTL. Thus, peace won't be brought until Robespierre is down or his ennemies are out of soldiers. Thus, instead of Napoleonic Wars, we would probably have Revolutionnary Wars that last around the same lenghth of time.
The ideals of the French Revolution are probably going to be further looked down with Robespierre staying in power longer since he applied them without caring for the consequences (in my eyes at least) and sending opponents to the Guillotine (that's what happened to the Monarchists, the Girondins, the Enragés/Hebertistes (pushed for a longer use of the terror) and the Indulgents (Danton and those who wished to reduce if not stop using the terror)). Thus, they will be less accepted than OTL and will take a longer time to rise up.