One is called the passive voice and focuses on the object of the sentence and the other is the active voice and focuses on the subject. Both are correct and you can use either one.

They would usually mean the same thing, but the one with 'got' makes it sound a bit less like it was your fault. It would be possible to say 'I mixed the days up' to suggest that you changed your usual routine to make things more interesting or something, but this would only be clear from the context.

Native English speakers use both and in context they are basically the same. For Example: 1. Mark: "Joey, you were supposed to come on Thursday, not today. " Joey: " Sorry, I got the days mixed up." 2. Ashley: " Did you forget my birthday is today?" Jake: "Oh no, I thought it was the 4th. I guess, I mixed up the days." You can say either to a native speaker and they won't notice a difference, nor will it sound strange to them.

They both have the same meaning...

"I got the days mixed up" - I was informed of the correct day of one or more events, but remembered/wrote it incorrectly "I mixed up the days" - there were two or more events and I had them remembered/written down on each other's days. Also FYI - in the past few years there has been an increased focus on using gendered second-person plural pronouns like "guys", and you might alienate some people by using it (some won't care). Some safer gender-neutral alternatives: "folks", "friends", "everyone"