Kent and The Fool actually share quite a bit in common. Both are loyal to King Lear and both are also loyal to Cordelia, albeit for different reasons. Kent logically sees the superficiality of Goneril and Regan but pleads with the king to see Cordelia's honesty. The Fool understands this reasoning as well, but his loyalty to Cordelia is also based on kinship with her, perhaps even love. When Lear calls for The Fool, a knight replies, "Since my young lady's going into France, sir, the fool hath much pined away" (I.iv.62-3). When Lear condemns Cordelia for failing to shower him with praises, it is Kent who sticks up for Cordelia and tries to reason with the king:
Thy youngest daughter does not love thee least;
Nor are those empty-hearted whose low sounds
Reverb no hollowness. (I.i.152-54)
Despite being banished by Learn, Kent returns disguised and becomes a (new to Lear) servant to the king. The Fool is also loyal to Lear. He and Kent try to give Lear honest advice about his predicament. But the Fool is more playful and sarcastic. In Act One, Scene Four, he gives Kent a fool's cap because he thinks he would be a fool to follow a king with no power over his kingdom. Kent is dubious at first but eventually agrees with the Fool's assessment of Lear's mistakes.
Kent goes to Regan on behalf of Lear. He actively tries to help Lear. The Fool, still loyal, plays a different role; often compared to the chorus in a Greek play. (The chorus would often comment to the audience what is occurring in the play. The chorus would give information that the other characters may be unaware of.) Kent is Lear's faithful friend and servant. The Fool is Lear's faithful friend and servant but he also acts as a prophet or a speaker of riddles, characteristic of the chorus. Kent is more direct in trying to get Lear to be reasonable. The Fool uses riddles and sarcasm to achieve the same ends.
This is an interesting question. The King likes the Fool because he amuses him and keeps him company. He is somebody to talk to. The Fool likes the King because Lear protects him and provides for him. Lear has led a life of privilege because he was a prince and heir apparent and has become the king. The Fool probably led a dog's life until he was picked up by Lear on a whim and made into a court jester. Consequently, Lear knows little about reality, while the Fool is worldly wise because he has been trained in the School of Hard Knocks. The Fool understands people because he has seen their dark side. He has even seen Lear's dark side more than once. Lear doesn't really understand people because he has always been flattered and treated with the greatest respect. Lear naturally believes that everybody loves and admires him. He finds out the sad truth that most people only care about themselves. The Fool doesn't have to find this out; he knew it from his birth. Ironically, Lear has more to learn from the Fool than the Fool has to learn from Lear. The Fool follows Lear out into the cold, cruel world because he loves him--but also because he has no other recourse. If the Fool were to remain behind he would be in grave danger. At best, he would only be thrown out into the storm. Even if Lear is no longer in a position to protect and provide for him, the Fool is better off with the King than he would be on his own. At the same time, the Fool can also offer Lear some good advice about how to get along in his new impoverished and homeless situation, because that is the kind of situation the Fool was very familiar with. Misery loves company. In almost any play, there have to be at least two characters talking to each other because all the information conveyed to the audience has to be conveyed through dialogue except when a character is soliloquizing. A good example and a good drama to use for comparison with King Lear is Waiting for Godot. The playwright Samuel Beckett had to have two characters waiting for Godot in order to be able to provide information, such as it is, via dialogue. So he created the memorable Estragon and Vladimir, whose plight is very similar to that of Lear and the Fool.