'Would like to' is most often used to say politely what you want or what someone else wants. 'I would like to go now.' It is also used to invite someone to do something. 'Would you like to have dinner with me?'
'Wish' can be synonymous with 'would like to' in expressing 'want' in a more formal way. 'You may look around, if you wish.' It is also used to express longing or desire for something that is unlikely to happen. 'I wish I were rich' and to say you feel sorry or disappointed about something you did or did not do. 'I wish I had never come'.