Question: I understand that one of them is actually the other. At first we are led to believe that Vincent is Ergo. But toward the end of the series, we discover something but I am not sure what it means: Ergo created Vincent to run away from himself. Also, in the beginning, when Vincent turns into Ergo, he loses control and doesn't remember any of it. Later on he starts to control Ergo, but at the same time he can talk with him.

So, basically, I don't understand which 'came first':
Is Vincent Law Ergo Proxy? Or is it the other way around? Or are they actually two different 'selves' that 'reside in the same body'?

Answer: 
From what I understand, Proxy 1 and Ergo Proxy are two separate entities. A miss conception, thanks to the English translation, is proxy 1 created Romdeau; this is not the case as Romdeau is described a flawed Dome in the same way as its creator, Ergo Proxy, is a clone on Proxy 1 and hence a flawed proxy (also shown by the fact he can survive UV rays). 

This is further backed up by the relationship between Ergo and Monad. Monad we are told loved Ergo proxy due to their opposing natures of life and death, thus she comes to Vincent AKA Ergos aid at the end of the series rather than to help forefill Proxy 1's ambitions for Ergo to destroy humanities chance for a future on earth.

Vincent is therefore the human name of the being Ergo proxy while he has amnesia due to the removal of his own memories to prevent him from becoming the literal grim reaper on earth. Ergo Proxy was created a long time prior to the series by Proxy 1 as a means of revenge on humanity.

Of coarse I could be completely wrong but it doesn't really matter as the main focus of the anime isn't the plot but the moral and theological questions which arise from the setting and premise of the series. In short its a fantastic anime whether the ending is ambiguous or not :)
According to wikipedia under [Other Characters - Proxies](https://en.wikipedia.org/wiki/List_of_Ergo_Proxy_characters#Proxies):

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> Ergo Proxy is a "clone" of Proxy One, Romdo's creator and guardian, who was created to help bring about the destruction of the human race because of Proxy One's anger at humanity's treatment of and plans for the Proxies, specifically Monad Proxy. Ergo Proxy often wears a white mask with elements of both The Phantom of the Opera and a harlequin jester to differentiate from Proxy One. Vincent initially has no control over his transformations, changing into Ergo Proxy whenever another Proxy reveals itself, but is implied to be in control of his abilities by the end of the series.
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Regarding Proxy One:

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> The main antagonist of the series, he is Ergo Proxy's original and true self, and calls Vincent his shadow. He was first alluded to in episode 15 and is the one behind the events of the entire series, having created Vincent and then sent him back to Romdo from Mosk to start his revenge plan. Near the end of the series, Proxy One is revealed to be the one who fired the thermonuclear missile Rapture, destroyed Amnesia to hide Vincent's memories, and killed Donov Mayer.
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To summarize, Vincent Law transforms into Ergo Proxy. Ergo Proxy is the proxy form of Vincent Law. He appoints Romdeau, his domed city, with a human regent and then gives himself amnesia, leaving himself in his human form (as Vincent). Ergo Proxy comes first, though he is a clone of Proxy One.