To understand this conflict you must first be aware that it isn't hastur and cthulhu which are hating each other, it's their minions.
There are also several other aspects that you need to know.
First of the mi-go is a race able to delete it's own memories individually, which makes the mi-go very hard to "read" as a species. Given that they do not have intuition their race is build on pure observation of trial and error, which means that they experiment and observe to learn about the universe. Such a society would sooner or later start experiments on itself, which may explain why mi-go's have warring fractions among themselves. A "control"-caste is able to decide different mi-go test civilisations and let several mi-go's learn the necessary and forget the rest. This is the direct reason why the mi-go's on yoggoth does not remember their origin in space, but still know that it's their duty to explore the sol-system.
The mi-go visited earth at a time where cthulhu was at war with the elder things. From their travels in space they still held the information about encountering both the elder things and the mythos, and since they had outposts on pluto it would benefit them to join one side or another. This would also give them valuable insight on both races.
The war amazingly ended with a truce where the elder things and the cthulhu split the earth in two parts. Cthulhu then went into hibernation.
Then the unthinkable happened. Humanity evolved and their first nuke told the mi-go that our mental makeup was special. They made experiments and found that humanity had evolved intuition, which bypasses research and observation in huge jumps. This is something that the mi-go want's for themselves, so humanity had to be preserved and the mi-go had to doublecross the mythos.
Cthulhu should have risen for some time, but the mi-go have changed the insides of our moon to control some of the tides on earth, and since the tides control the wakeup sequence of cthulhu, he remains asleep. The mi-go's are also having dimensional dampeners in their lairs that prevent Hastur from manifesting on earth, which is why the brothers of the yellow sign" is at war with the mi-go, trying to torture information out of them (Hard considering their memory deletion capabilities) The mi-go keeps the rest of the mythos at bay by claiming that it would be heresy to let hastur roam the earth without cthulhu, since they are halfbrothers and both hold the birthright to guide the armies of the mythos on earth.
This is the current situation :D