The infinitive ‘to win’ implies chance or competition playing an important part in gaining a prize or position. The infinitive ‘to earn’ implies that something has been done, perhaps a service performed, behavioural pattern or work carried out, in order to receive an entitlement to something e.g. a position, money or trust.

Win means to complete a thing done before anyone else. Earn means to get something in exchange for your service. Like earning money in exchange for your work.