This turned out to be much harder than I thought it would be. I thought I knew what I was talking about, but while researching I found out that I really don't.
There are two things here: the way to connect the die (the silicon chip) to the package, and the way to connect the package to the PCB or whatever you want to connect it to.
Flip-chip refers specifically to "C4 flip-chip" - which is a technique and a chemical process to connect a silicon die to a package. BGA is a package - it's uses solder bumps to connect a package to another component (like a PCB).
The Pentium III uses FCPGA - which is a packaging technology that takes a die, flips it and connects it directly to a PGA package. The K6 and Cyrix 6x86 PR300 used a form of BGA to connect the die to a BGA package and then connect this BGA package to a PGA package (since BGA in non-removeable, so once you solder it you can't remove it, and people like to upgrade).
A big differences between FC C4 and BGA is that within the BGA package there are wires that connect the die to the outside of the package whereas in C4, there are just bumps that connect the die directly to the package. In BGA you only wire to the periphery of the die (the outside edge) and so you have a limited number of wires (those that can fit on the outside edge) and these need to be routed across the die to wherever you want them to go. With C4 FC, you put bump spots all over the die and so you are no longer stuck using only the periphery, you can connect everywhere. So rather than having, say, 300 wires, you can no have thousands. With C4, the die directly contacts the heat-spreader (or the back of the heatsink). With BGA, the die is encapsulated within another package.
So, C4 is better. But it's much harder to do and it's a newer technology. It took years to get it working. The Athlons are all C4 FC, but the K6's and 6x86's were on some form of BGA (micro-BGA, etc.).
A cut-away picture of a
BGA package.
A side-view picture of a
microBGA and easyBGA package.
An
article on Intel's FCPGA package technology.