What is the difference between the "affine connection" (S. Weinberg, Cosmology) and "Christoffel symbols?"
They are very closely related- so much so that Christoffel symbols are commonly also called "Connection coefficients." In a curved space, comparing one vector (or other mathematical object- tensor, n-forms, etc.) to another is not so straightforward a task as it is in nice, flat, Euclidean space. Misner, Thorne, and Wheeler's textbook Gravitation really works out the concepts in extreme detail to make them clear. Basically, you need to calculate some corrections when differentiating in a curved space or else you will get anomalous answers that depend on the details of your calculation.
The affine connection is the conceptual link between two very nearby points where the vectors you would like to compare reside. The Christoffel symbols are the means of correcting your flat-space, naive differentiation to account for the curvature of the space in which you're doing your calculations, between those two points. So you could even call the Christoffel symbols "the same thing" as the affine connection, in a sense similar to calling a vector and its components in some particular coordinate system "the same thing."
Let me elaborate a little on Andrew's answer, and perhaps provide a slightly more mathematical perspective.
In the setup of curved spaces (that is, manifolds), one usually regards the vectors originating at a point $p$ as completely different from the vectors that originate at a point $q$. Said another way, to each point in the curved space, we attach an entire vector space full of vectors (called the tangent space at the point). The catch is that the tangent space attached to a point $p$ may have nothing to do with the tangent space attached to a different point $q$.
This is where the notion of an affine connection comes in. As Wikipedia and Andrew both point out, an affine connection can be used to "connect" vectors that live at different points. This process is called parallel transport. Intuitively, this involves sliding a vector along a "straight line" (geodesic).
Of course, since the given curved space might be really curved, the "straight line" might not look like a straight line in the flat euclidean space that we are used to. For example, the straight lines on the surface of the sphere are really great circles, like the equator.
All of that being said, let me answer your actual question.
Loosely speaking, an affine connection is really a function that inputs vector fields and outputs vector fields and satisfies certain rules (that I will not go into here). A given space may have more than one affine connection, but we usually like to pick a specific one called the Levi-Civita connection.
Now there is one more piece of terminology that is needed here, and that is the notion of a moving frame. Essentially, a moving frame is a choice of basis for each tangent space (which is often required to "vary continuously" in some sense).
So, given an affine connection and a choice of moving frame, the Christoffel symbols are in some sense the components of the affine connection with respect to the basis determined by the frame. In other words, the affine connection -- together with a moving frame -- determine the Christoffel symbols. As Andrew notes, this is just like how given a vector, we can determine the components of that vector with respect to a basis.
Conversely, given collection of Christoffel symbols, one can actually construct an affine connection for which those are the components with respect to some local frame. Again, the analogy to vectors and bases works here, too.