Thank you. The first thing to note is that as you did not accept the offer, there would have been no contract formed and as such you cannot claim that they were guilty of breach of contract by giving the job to someone else. Your rights will be limited to any protection you get under equality laws, which deal with discrimination. For that, you must be able to show that you were treated detrimentally because of a protected characteristic, such as age, gender, race religion, sexual orientation, disability.
They mentioned cultural differences. This does not automatically man something related to your race or religion. However, if you believe that it was to do with any of these characteristics, then you can consider a claim for discrimination against them but you would need some evidence to back it up. Only in that case can you consider challenging their decision, otherwise with no formal contract in place they can refuse to employ you at any time.
This is your basic legal position. I have more detailed advice for you in terms of the options you have if you think that there was discrimination at play, which I wish to discuss so please take a second to leave a positive rating for the service so far (by selecting 3, 4 or 5 stars) and I can continue with that and answer any further questions you may have. Don’t worry, there is no extra cost and leaving a rating will not close the question and we can continue this discussion. Thank you