Anger is great displeasure and covers many degrees of this feeling.
Rage is extreme, uncontrolled anger.
"He flew into a rage when he found out about his friend's betrayal."
Wrath is an old form of rage and is not used much, except to emphasize a point or make it somewhat humorous
Resent is used when there is anger which is not open so much. When we talk about outward displays, it is usually pointed out.
"He resented being passed over for promotion."
"He express open resentment at being passed over."
"He faced his father's wrath when he came home late with the car."