>>> Call me thick but I don't understand the difference between a proper email client and a web-based one
With a web-based service you need to be online all of the time that you're doing anything with your email because your emails are stored on the website of the provider that you're signed up with.
With an email client your emails are stored locally on your computer. So, say, to read all of the mail that you've received in the past few days you don't need to be connected to your email provider's server.
If you want to check for new mail you simply click a button and (assuming that you've opted to have your account login and password stored in your email client) you get connected to the server, with all new mail then downloaded to your client, without the need to manually sign in.
In fact you don't even need to click a button, as you can tell your email client to automatically check for new mail periodically. (That can be every second, every minute, every hour or whatever other time interval you choose).
When you're writing an email you're not connected to your provider's server (so it's impossible for you to be logged out, as you were never logged in anyway). You can take as long as you like to write your message, without fear of it disappearing into cyberspace. When you're ready to send it, clicking on 'Send' automatically connects you to your provider's server (with no need to log in) in order to achieve your aim, before disconnecting you again.