"Let's check another store" means "Let's take a look at another store" "Let's check out another store" can mean the same as the above, or "Let's take a closer look at another store". There's more a sense of taking a closer look or examination with the phrasal verb 'check out'. In your example, I prefer 'check out' as as it emphasizes a little more that you want to take a closer look at this other store.

Check means to examine Check (n) = A written order to a bank for payment. Check out = Leave/vacate the hotel/lodge after paying the balance. ( or consult/reassess )

Hi Sonia, I think in this situation they are more or less the same. If I had to specify a difference between them: “Let’s check another store” : definitively suggests that you are searching for something (the perfect pair of shoes to go with your outfit) “Let’s check out another store” : might mean you are looking for something or it could mean that you are just interested in looking around another store (maybe you’re bored, or the music is too loud, etc) In many situations you could use either, and people will understand what you mean depending on the context.