The rivalry between Athena and Poseidon is not only well known, but legendary. This has been going on since the two gods were fighting over Athens. I am unsure of what the city used to be called, but it is surrounded by the Attica region. Both gods wanted to become the patron so the people had them both create a gift for the city. The king and people of the city went to the acropolis for the competition.
Poseidon (God of the Seas) struck the earth with his trident(a three pointed spear) and a created a spring. The people loved it at first, but as they tasted the water they discovered that it was too salty to drink. Salt water was not useful to the people.
Athena (Goddess of Wisdom, battle strategy, arts, and literature) planted a branch. This branch grew into an olive tree. This was much more useful to the people as it could be used for food, turned into oil for their lamps and cooking, and the wood could be used to build homes and boats. The olive tree is also a symbol of peace and prosperity. The people chose Athena as their patron, as her gift was more useful to them.
The fight for patronage was not the only conflict between the two gods. There is also the change of Medusa. This is another well known story, however like all, myths there are multiple versions.
Some say that she boasted how she was more beautiful than Athena. Another says that she lived far north and requested Athena to go south. Athena denied and Medusa called her jealous of her beauty in response. As a result Athena not only made her hideous, but cursed her so anybody that looked at her turned to stone. The one most people know is that Medusa was so beautiful that she was courted by all men, however she became a priestess of Athena. Her beauty attracted the God of the Seas, Poseidon. They slept together in the temple and as a result she got pregnant. Some people say that Medusa seduced Poseidon and others say that she was raped. The result was the same either way. When Athena discovered what had happened in her temple she was furious. She cursed Medusa with a hair of snakes, body of a dragon(could have been a snake), and ability to turn whoever she looked at to stone. When Poseidon looked upon her he was disgusted and left her. Eventually a son of Zeus,Perseus, killed Medusa by cutting of her head. From her blood came out two children of Poseidon, Pegasus and Chrysaor.
There is much animosity between Athena and Poseidon because of these two issues. I am sure there are many more, but these are the two that I know the most.