What’s the Deal With Sending Emails From Abroad on a Windows Client?

Here’s a question for the technically minded among you.
Back in olden days (1999 or so), I always had trouble with email when I was on the road. I could receive but I couldn’t send, both in the US and abroad. My only choice was to use my internet provider’s ghastly web-based app for sending email. Web apps and built-in smartphone apps always seem to work without problems, but my normal client (Outlook at the time) was useless.
In any case, that’s in the dim past and I’d forgotten all about it. Then a friend of mine went on vacation to Ireland, and when he came back he complained that he couldn’t send emails from there. He spent an hour on the phone with Cox, our internet provider, until they evidently flipped a switch or something and he could once again send emails.
When I got to Ireland, I figured I might have the same problem, but I didn’t. I’m on T-Mobile, and it connected as soon as we landed. I sent and received email just fine on my phone. Then we got to our house in Kerry and I connected my tablet to WiFi. No problems. The Windows client worked fine. I set up the phone as a hotspot and connected to that. No problems. We flew to London and I connected to the WiFi in our house there. No problems. One way or another, I figured that Cox had changed its tune.
Then we flew back to Dublin for a few days and I sent some emails. At the airport on the last day I sent another one. But then I noticed that they were all still pending. It turns out I couldn’t send emails via our hotel’s WiFi or via the airport’s WiFi. Apparently the ability of a normal Windows email client¹ to send email depends on exactly what server I’m connected to.
What’s the deal with this? Since I can always send email by connecting to my phone’s hotspot, this is no longer too big a deal. But I’m curious. Why would I be able to send emails from some places in Ireland and Britain but not others?²
¹Mine happens to be eM Client, but every other client I’ve used acts the same way.
²I’d call Cox and ask them, but I assume that would be a waste of time. I’m looking for a real answer here, not marketing gibberish.