Neither phrasal verbs nor regular verbs with the same meaning are "better". It depends on the situation. In general, phrasal verbs are used more in informal speech. This is especially true when the action is physical. Would you please get out of here? (not "leave") Please take off your clothes. (not "remove") However, in more academic situations, a lot of phrasal verb are used over the other verb. They carried out the experiment over a period of three days. (not "performed" or "did") She pointed out the flaws in the data. (not "indicated" ) In your examples it really depends on the formality of the situation. "Don't disappoint me." (a boss to his sales representative) "Don't let me down." (one guy needs his friend to show up on Saturday to help him move into his apartment) "I'm sick of all this arguing. Let's just call the damn party off." (informal) "Can I have everyone's attention, please? We need to cancel the party because the administration says it's against policy to have parties in the classrooms." (more formal)

For the first pair: I hear "I want to call the party off." or "should we call the party off?" more often. Second pair: I hear "Don't let me down." more often

Dark Panther gave you an excellent answer for the first part of your question. . Don't disappoint me' is usually said by parents to their children 'Don't let me down' is often said by employers to their employees.

'I want to cancel that party' is more common I think.

They are the same. It is irrelevant which one is more common.