I've had this scots pine since winter, and a few months ago I started noticing white fluffy bits at the base of the needles. It looks like wooly aphids to me. The needles are turning brown and falling off and some branches are becoming brittle enough to snap off. Can anyone confirm the problem and how to save the tree?

Not sure that you have aphids, from the pictures. It could be dried sap too.

But even if they are, I doubt they are the cause of the problem.
You said you got is recently. DO you know whether this tree was repot in the last 6 months? It looks more like rootfailure to me.

If you rub it off is there an insect underneath? I tend to agree with LB., but I use a mixture of a few drops of washing up liquid and a small amount of vegetable oil say teaspoon in a litre of water and spray it.

I potted it in late February when the weather warmed up. It was in a small plastic nursery pot and extremely pot bound. I teased out the roots but didn't really trim much. The roots on the outside seemed a bit dry.

I can't see any insects with my naked eye, but the fluff looked like pictures of wooly aphids. It might be a fungus, but I wouldn't know which. I'm in zone 9 in California if that helps.