From the information I can find online, the only real difference between the two is the remote. The 3821 appears to come with the Voice Remote Pro, which has the rechargeable battery. The Wiki table shows identical specs for everything else. You can use any Roku remote other than the basic IR remote. The Roku Stick players do not support IR control.
You may encounter an issue when traveling to a different country. Your app/channel access is dependent on the country your Roku account was created in and where you are physically located at when using the device. If your account was created when you were on a computer located within the US, you will only have access to channels/apps that are available to US residents. Simply taking your Roku to another country will not allow you access to apps that are specific to that country. It will also mean that you might not be able to use all of your US based apps while located in another country.
To use your example of being in the UK, you will not be able to install the BBC app and use it while you are there. The BBC restricts what devices are permitted to use their service by model number, so only UK specific models will work. That doesn't mean your Roku will be useless while there, as there are some services that don't care where you are (such as YouTube from what I understand).