Rather/Prefer

Rather and Prefer are words used to show a choice from different options.
For example:

• Would you rather eat pizza or pasta?
  
• I would rather eat pizza.

• Would you prefer to eat pizza or pasta?

• I would prefer to eat pizza.

STRUCTURE
The structure of these sentences are the same except for one small difference:Would Rather is followed by a base verb and Would Prefer is
followed by an infinitive.

Subject  +  Would  +  Rather  +  Base Verb

• I would rather live in a warm country.

• I would rather go to a different restaurant.

Subject  +  Would  +  Prefer  +  To  +  Verb

• I would prefer to live in a warm country.

• She would prefer to meet on Monday.

Not comes before the verb with both rather and prefer.

• I would rather not go to work today.

• I would prefer not to play a game.

Take a look at this conversation to see some examples of would rather and would prefer:
Alex: 

Would you rather stay home and cook dinner or go out? 

Betty:            
I’d rather go out.  I am pretty tired today, so I don;t want to
cook.  What about you?  Would you rather stay in or go out? 

Alex:              
I’d prefer to stay in, but we can go out if you would like.

 Betty:                 
That would be nice.  What type of restaurant would you like to go to? 

Alex:               
I’d prefer to go somewhere quiet. 

Betty:               
Really?  Not me.  I’d rather go somewhere fun, maybe even with some live music. 

Alex:               
But I thought you were tired?

Betty:               
I am, but not too tired to have some fun.

Alex:               
Well, I’d prefer not to, but…..OK.