They are both correct. As Sergey said, 2 is a more formal way to phrase it. People who use the Prep+whom combination are considered to sound more "high brow" (ie. they sound fancier). However grammatically they are both correct.

In formal English we use preposition + whom: With whom did you go? To whom were you speaking? But in ordinary English we usually move the preposition to the end of the sentence. The "whom" then normally changes to "who": Who did you go with? Who were you speaking to?

I can only guess that sentence number one is more formal and rather used in written form. Whereas the second one is more likely in spoken context.

Hello Liza. I think it's better to be written as: 1. Who would you like to spend your vacation with? 2. With who would you like to spend your vacation?

Hello Liza. I think it's better to be written as: 1. Who would you like to spend your vacation with? 2. With who would you like to spend your vacation?