To put it simply, they were the same people.
According to Mark C. Elliot's The Manchu Way: The Eight Banners and Ethnic Identity in Late Imperial China, 47-48:
In the search for Manchu origins, the most important point to note is that in the beginning, there was no Manchu nation, only many Jurchen tribes. The name "Manchu" and the creation of a unified state came about a virtually the same time, in the 1630s, prior to which time it is correct only to speak of Jurchen (Nvzhen/Jusen) peoples, Jurchen tribes, Jurchen leaders, and Jurchen institutions....
....The core of what would become the "Manchu" people - that is, the people at the center of the banner system - were originally either Jianzhou or Haixi Jurchens, along with two of the Yeren tribes; the rest of the Yeren were never integrated, socially, politically, or economically, into what later became Manchu structures.
It's worth noting that prior to the establishment of the Qing, Nurhaci was the Khan of the Later Jin (後金), which he claimed to be the successor state to the Jurchen Jin dynasty.
On page 63, Elliot talks about the creation of the Manchu identity under Hong Taiji:
These efforts at consolidation came to a climax in the mid-1630s, a major watershed in the development of the Jurchen state. First, the name "Manchu" was officially adopted in 1635 as the name for all Jurchen people.
The use of the term "Manchu" served to forge a new pan-Jurchen identity. This new identity was used to unite the divided Jurchen tribes into a single entity to combat the Ming.
Lordtiandao's answer is good, but I'll just add that Manchurians generally consider themselves to be descended from Jurchens, but not exactly the same.
I think an equivalent would be that English people consider themselves to be descended from Anglo-Saxons, but if we talk about "The Anglo-Saxons did this" we don't generally mean something that the English did last year, we mean in the 600s.
(At the risk of personal anecdotes, I actually live in Manchuria and have discussed this a little with some Manchu)