There's numerous potential reasons why they went extinct. The main one is that they merely couldn't adapt fast enough after the ice ages passed and were out-competed for food, however it might have been that humans basically killed the Neanderthals through the spread of farming.
Homo Sapiens Sapiens (us) are keen users of fire throughout history, in fact we made the world we chose to live in. Through purposeful setting of forest fires we converted forest to savanna, which we could then use for farming and pasture.
The interesting thing though is that the first (fossil recorded) domestication of dogs happened in 30,000 BC in Belgium, and likely this behavior had been going on for some time before a dog conveniently died in a way to be fossilized and not used for food. Carbon dating suggests Neanderthals went extinct 30,000 years ago (from the latest remains, however again some will have lived longer without being fossilized), which is in fact 28,000 BC. So the question is, did Neanderthals begin the domestication of dogs? Because the next known domestication occurred around 7,000 BC again in Belgium.
So I wonder, did Neanderthals show humans how to domesticate dogs? Because throughout human history, invention spreads quickly, so why did it take almost 25,000 years for it to provably occur again in the same place?
The question of intelligence though depends on a number of factors. The human brain is about as large as you can get before signal delay gets too long (possibly a reason for Neanderthal extinction, they literally couldn't react fast enough), however the human brain has a high nerve density, which would just compound the problem if our brains were bigger. It's certainly possible that the Neanderthals were as intelligent as us, but that they simply reacted slower and when the environment started warming up and more dangerous animals started spreading back they might not have coped as readily.
An extreme example of the neurone delay is the Elephant. It's brain is massive, but they have relatively slow reactions. However their intelligence is immense for an animal, I mean it appears that they actually have the concept of revenge, which suggests an understanding of right and wrong.