If I have budget for one of two graphics cards as an upgrade, where one has more RAM but the other a faster clock speed on the GPU, which would have the biggest impact on gaming performance? The one with more RAM, or the "faster" one? All other factors should be considered equal (drivers, chipset, manufacturer, price, etc).
You need RAM to hold the data - polygons, textures etc. More ram will allow you to show more detailed textures and run at higher resolutions.
You need a good GPU to do all the calculations needed for a modern game - lighting, physics etc. The better the GPU the more of these calculations you can do so you'll get better effects and it will have an impact on the resolution too.
Check the specs of those cards and then you need to decide what's important - resolution, effects etc. and go for the card that's better at those things.
The real answer is neither:
A high GPU clock speed means nothing, especially when you are comparing against brands. It is not on how many clocks per second a core runs at but rather how efficient the core is. For example, a GTX Titan has a core clock of 836MHz whereas the HD7770 has a core speed of 1000-1100GHz, so based on clock speed, the HD7770 is clearly better right?
Wrong. The Titan is actually 4-6 times better than the HD7770 because its core is more efficient. You will have to look at gaming benchmarks to figure out GPU core superiority.
Next is the RAM. Pure RAM numbers tell you absolutely nothing about how a card will perform. A large amount of RAM will not always help you play games at higher resolutions, especially when a large amount of RAM is paired up with a really inefficient GPU (notice I didn't say low clocked GPU). Only powerful and efficient GPUs will be able to take advantage of 3Gb-6Gb of memory in a card.
Just like a computer, a card's components can bottleneck it. An HD7770 with 2 Gb of RAM is a waste because the GPU core cannot handle 2Gb of data at a time while a Titan with 6Gb will be able to use most of its memory efficiently as its core is able to handle 6 gigs of texture data.
This brings me to the last topic mentioned, high resolution. More memory will not help you play games at higher resolutions but rather, are a component that can. Like I said before, you need a very good graphics core paired with large memory in order to get the best experience. The memory must also have a large bus. 2 gigs of RAM in an HD7770 is a massive waste not only because of its slow GPU, but also because of its 128-bit bus. You have to have a wide bus in order to have higher data transfer rates between the RAM and the core. The 6 gigs in a Titan can travel in a 384-bit bus, which is sufficient in transferring data between the core and the memory.
Final Tip: When shopping for a GPU, compare cards based on gaming benchmarks and not hardware specifications (except for the memory bus size ;)).
Having a big amount of RAM could help when playing at big resolutions, let me say
1920x1080 and more.
For resolutions like
1680x1050 or less 1GB would be great but even 512MB is enough for most of games event at the highest texture detail setting.
If your monitor is
1680x1050 or less and the video card you choose has already 1GB I would suggest you to go for faster clock speed otherwise take the one with 1.5GB or 2GB of RAM.
ps. Clock frequencies can be overclocked, RAM amount cannot be increased ;)