Some animals, like whales and elephants for example, (probably–?) have bigger brains than humans. Yet we are clearly vastly superior to them in intelligence. Why are their bigger brains somehow less sophisticated than ours? And indeed do their bigger brains give them any advantage over us? Do they, for example, have greater depth to their emotions 
?
Well, they’re better swimmers…
There’s really no correlation between brain size and intelligence.
The ratio is more important than the actual size. Whales are bigger animals, but in porportion, their brains are smaller. Secondly, they use their brain to do different tasks than we do.
This site has details on why bigger isn’t better for whales. They say:
As Kimera said, size isn’t important, it’s the relative size that’s meaningful. Brains perform all sorts of functions but the single biggest task performed is simply controlling the body. Even in humans IIRC something like 75% of the brain seems to be dedicated exclusively or primarily to controlling the body. That control takes the form of telling the muscles what to do, interpreting input from the environment such as pain and temperature and simply keeping track of where all the various bits are in relation to each other.
The logical extension of that bit of knowledge about the brain is that bigger animals will need bigger brains. Simply to interpret all the data coming in from what is probably a hundred square yards of skin requires a huge brain for a whale or elephant. Humans are much smaller and require far smaller brains just to control the body.
However the human brain is massive compared to our body size. For example it’s larger than the brain of a horse, an animal that is several times larger. It’s also larger than that of a gorilla, which is also a much larger animal. The brain to body mass ratio of humans is larger than for any other animal.
The same is not true of elephants or whales. While elephants and some of the dolphins have brains that are slightly outside the expected range for their body mass even that isn’t true for the baleen whales, which tend to have brains that are either normal size for the size of the animal or even on the low side.
What that means is that after dedicating the required amount of brain power to just staying alive these animals don’t have all that much to play around with for abstract thought or problem solving or emotions. Humans on the other hand have what amounts to a spare 25% of brain mass just to dedicate to those sorts of things.
Can we see a reference for that claim please? Everything I’ve read has shown a reasonable correlation between gross brain size and intelligence (dogs are smarter than cats are smarter than rats etc) and a very, very good correlation between brian:body mass ratio and intelligence.
Wow, great answer, Blake. I learned a lot.
Proportion is important, but I believe intelligence is also related to the amount of surface area on the brain, and within the brain. In general, the more convoluted (or noodley) the brain, the smarter you are. The surface area is covered with gray matter, underlain by white matter. The amount of gray matter may be very important, i cant remember…
Yes, grey matter is very important to intelligence. For example, on average, women have smaller brains than men but more grey matter.
Blake, Smeghead might be refering to the belief that within a species, size is more important than ratio and that individuals with larger brains are more intelligent. There was a weak correlation found before that suggested individuals with larger brains were more intelligent, but a lot of new research is suggesting that size doesn’t matter as much as grey matter.
Dr. Haier at the University of California Irvine did a study where they found that the amount of grey matter plays more of a difference in intelligence than the actual size of the brain.
This google answer contains a lot more interesting information and links. It includes a list of ratios of brain size amoung several animals.
Brain Weight as % of body weight.
Man 2,1
Asian elephant 0,15
Killer whale 0,094
Pilot whale 0,076
Sperm whale 0,021
Fin whale 0,008
Aren’t you making an assumption here? 
Wouldn’t a large part of a dolphin or whale’s brain be dedicated to processing sound? How much of the human brain is dedicated to processing visual information?