HTML website stops working as expected after migrating servers

I'm a web developer for years and I understand a lot about websites and development. But this issue is far from my understanding, I had never encountered it before as it looks mystical for me.

I have HTML website on main server and domain, it's working fine there, all files are loaded and design looks as it should. Now, I have moved the same website, all the same files to a new server and domain, and the website just broke without touching anything. It's working, but design doesn't look like it should, Javascript is throwing errors in console, design structure is lost. All files needed are loaded, I can see in the Network tab of Chrome Developer Tools but website design structure is lost and JS scripts are not working. Now this is the fun part. If I transfer the same website from new server to the main server, the website looks good once again. I'm stretching my head all day and can't find any issue. I've also tried to run the same website locally on WAMP or even just from desktop as it's only HTML, and the website is broke as it's in the new server.

Main website URL: http://vilbar.lt/eng/ New server's website URL: http://www.wellsources.cn/eng/

Any ideas? Anyone had encountered this before?