What client do you use to connect to Hotmail on the laptop. MS made some changes back when they migrated users to Outlook.com (essentially Hotmail, but not exactly). I think this would have been apparent a while ago, but you never know.
There are also some new account security features (although not necessarily that new).
And then ... there's the Outlook Connector (I'm not familiar with that at all, but it might be part of the equation)
Suggestion: Log into your Hotmail account on the web and check the configuration there. If there are any 'verify my account' links, try those. If there are additional security features, look through those and see how they might apply to the client you use.
The fact that different devices connect doesn't help troubleshoot the issue on your laptop, other than prove that it's not a router issue - that's a good thing. Thanks for that information. Those devices use different clients no doubt.
There's always the option of changing the protocol from HTML to Imap or POP3 - but save that for last (some research is required).
That's all I can think of given the information I have. You could preempt my next request by posting the configuration on your client.
Be sure to clear anything sensitive from the information you post, such as your eMail address (double check before posting ... sometimes the eMail address is also shown in the title bar of a window).
Bill
.