What is the difference between mutually exclusive and indepedent events?

Let's say that we have two events and we'll call them event A and event B. If these events were mutually exclusive then these events would not be able to occur at the same time. For example, if Real Madrid and Barcelona were playing each other in a football match, event A would be the event in which Real Madrid win the game and event B would be the event in which Barcelona win the game, these events are therefore said to be mutually exclusive as only one of these events could happen, it is impossible for both events to occur at the same time. For indepedent events, both occurences are completely unrelated and have no influence over each other and can both happen at the same time. For example, let event A be the event in which a boy buys an ice-cream and event B be the event in which it rains in Mexico. These events have no influence over each other whatsoever as rain in Mexico would not encourage/discourage a boy to buy an ice-cream and a boy buying an ice cream would not increase/decrease the probability of rain in Mexico and of course these events could both happen at the same time, i.e. they are independent of each other.