Mdthbs

I read this article and it seems like activists from cultures without a long history of whaling are inconsistent in their criticism of Japan for its whaling.

Many cultural cuisines like rabbits, cows, trout, pigs, chicken, reindeer or walruses are consumed without international criticism. Why does the international community strongly condemn whaling and not other forms of fishing and hunting? I'm looking for the stated reasons from environmental groups and governments that have signed the the International Convention for the Regulation of Whaling.

I think there are two main reasons. Ethics (which does apply to dogs in many parts of the world) and the fact that there aren't that many whales (as opposed to dogs).

Ethics

It's seen as immoral. This arguments also applies to dogs. As such, you see a lot of outrage about dog consumption as well.

For example, these quotes from a BBC article on dog consumption in Vietnam illustrate this point:

The Hanoi People's Committee said the practice could tarnish the city's image as a "civilised and modern capital".

[...]

A growing number of people in Vietnam disapprove of eating dog meat but it still remains "very much a deep-rooted habit", according to Linh Nguyen, a journalist with the BBC's Vietnamese service.

Threatened species

The second argument doesn't apply to dogs: there aren't that many left. This raises conservationist issues as many people and countries don't think it's worth making (some species) of whales extinct just for some delicatessen.

Specifically, the whale is an endangered species, meaning (from Wikipedia):

a species which has been categorized as very likely to become extinct in the near future. Endangered (EN), as categorized by the International Union for Conservation of Nature (IUCN) Red List, is the second most severe conservation status for wild populations in the IUCN's schema after Critically Endangered (CR).

The WWF provides some info on why whales are important for ocean life:

Whales are at the top of the food chain and have an important role in the overall health of the marine environment. Unfortunately their large size and mythical aura does not protect them; six out of the 13 great whale species are classified as endangered, even after decades of protection.

What is the fundamental difference between catching whales and hunting other animals?

Culture

There is no fundamental difference between whales, and e.g. dogs or cows. All are mammals; Some people eat some (or all) of those, while some other people consider it immoral to eat some (or all) of those.

Therefore, the main difference is culture. Many people are used to eat beef, while they are not used to eating whale meat.

I'm only going to try to cover things that haven't been mentioned in other answers and try to preempt any begging of the question that might come.

There are environmental and practical concerns to eating certain animals that don't apply to the same degree as other animals.

For example, pretty much all meat provided by domesticated animals come from prey animals, this means that less energy is lost as they are grown since they're lower on the trophic level. Meat that is "harvested" from the wild tends to be relegated to commercial fishing, where the large majority of fish is again, not placed at the top of the food chain. From this perspective, eating chicken is superior to eating beef, and eating anchovies is superior to eating swordfish.

Whales are either apex predators or filter feeders that have no natural predators. They are as a group the physically largest animals that exist on Earth. Eating them is highly inefficient from the perspective of obtaining cheap efficient calories (you basically can't get worse than eating whale). Allowing only market forces to regulate this (I expect whale meat would be quite expensive), is not enough since the externalities involved in losing large amounts of whales would not be calculated into their sale.

Since there is already an answer stating that only cultural differences apply when choosing among a variety of meat animals, I will point out another problem (though it doesn't really apply very well to whales in particular). The consumption of certain animals are more dangerous when it comes to public health, generally this applies to animals more closely related to the predator, which is why you see things like Ebola breakouts in Africa, where people are still in contact with various large primates (and occasionally eats them). Following the logic that all meat is equal would mean that something like the ritualized cannibalism of dead relatives is equivalent to eating pork (cannibalism might even be morally superior under various ethical frameworks assuming you didn't kill your relative and did kill the pig). However human flesh is statistically much deadlier than pork and it's probably a bad idea to engage in cannibalism when you have safer sources of food around. To relate this Japan it would be better to look at the sale of the Japanese Pufferfish, which is either banned or highly regulated across the world because its body is full of neurotoxins. The fact that there is still private demand for something that can be considered a "cultural cuisine", doesn't preempt government regulation or mean that the regulation is attempt to impose foreign values on native cultures.