This depends somewhat on the purpose and complexity of the website, as well as where most of the customers are located.
Assuming the customers are located in the US, hosting the site in the US will provide a lower latency and likely higher bandwidth. This could shave (roughly) about 100ms of the latency - this is not massive, but is more then implied because loading a web page requires multiple round trips (eg a minimum of 1 for SSL negotiation - although often more, 1 for the web page, and 1 for the images - so if the latency difference is 100ms, the minimum page load time goes up by 300ms by hosting it in Europe). You being in Europe would likely give you a different website experience to people in USA
Also, generally the closer to the client, the greater the bandwidth available. This is often due to better connectivity for closer devices, but also because tcp/ip takes latency into account to determine the maximum speed. If the site is highly optimised this may not be a big deal though - but for sites with large images it can be significant.
I am very biased, but I perceive Hostgator as an average provider - middle of the range performance at a good price. In the "middle-of-the-range" providers at these both the above pricepoints you are signalling that performance is of secondary concern - and it is quite possible the cheaper European hosting will outperform the Hostgator pricing (as an anecdote, I have servers in New Zealand, and host a number of sites for companies in Asia. My servers performance often outshines closer providers even though there is an extra 150ms latency - but my servers are way more powerful and optimised - and the client sites benefit from the extra CPU)