Information Technology Specialist here.. what you are describing is private IP vs public IP, private IP addresses for most routers / modems will be 192.168.0.1 or the last two octets may be 1.1 or 2.1 for example. This is standard for MOST routers, and these are your private IP addresses. The reason for this, is if every device in the world had its own public IP addresses, there wouldn’t be enough to go around with the current version of Internet Protocol (IPv4) which is why IPv6 was invented, but that is not fully implemented and it is in hexadecimal format. If you’re banned from Twitter, it is likely an IP ban, changing your private IP address will not fix your problem because Twitter will only see your public IP. A good way to test this is to go to a website like IPChicken.com (this will give you your public IP address. Write it down, then go to windows, open command prompt and type IPconfig. If your numbers are not the same then you are using a private IP address that is being converted to public.
I would not recommend using a VPN, it sounds like you may have gotten lucky and had a good server close to you, but it will always be slower as your network traffic is being routed elsewhere (along with everyone else who uses the service) processed, then sent to where it needs to go.
If you are banned by MAC address rather than IP, why not just buy a new $20 network card? This is where your MAC is tied to, and it is unique to all cards. All you have to do is unscrew the current one from your PCiE port and pop it out, replace it, new MAC. Or try downloading a MAC address spoofer, but they can be a little tricky.