Switch to Gmail: This is your best long term solution. MACMac OS X 10.6 Snow Leopard and MACMac OS X 10.7 Lion properly supports Gmail/Google Calendar/Google Contacts.
Use mBox Mail (with any Mac e-mail app that supports IMAP): It's been updated to support MAC OS X 10.6 Snow Leopard and comments above says MAC OS X 10.7 Lion too. It was lasted updated in August 2011. This won't sync calendar/contacts just e-mail/e-mail subfolders. It is only a one off fee of $19.99 USD.
Use IzyMail (with any Mac e-mail app that supports IMAP): I believe this works in a similar way to mBox Mail, by way of a web proxy to the companies own IMAP server. This also won't sync calendar/contacts just e-mail/e-mail subfolders. The website looks dodgy to me but it is only $18 USD a year.
Use Hotmail.com (or Gmail.com) and forget the idea of using a desktop client. This wouldn't suit me but it does many people.
Use Windows. I know hardly a solution but Outlook 2003-2010 (after you've installed the Hotmail Connector add-on) and the free Windows Live Mail fully support Hotmail's DeltaSync. Which is ultimately what you need to get what you're after.
Even using mBox Mail/IzyMail you still won't have your Calendars and Contacts synchronised. As far as I know, there's no way to get these into MAC OS X software (HotmailHotmail doesn't support CalDAV which is the universal calendar equivalent of IMAP), meaning no Mac OS X software is able to talk to Hotmail's calendar and you can forget contacts sync.
All the other options mentioned by others above look very complicated, certainly more so than switching mail accounts (assuming mBox Mail and IzyMail fail).
Migrating from Hotmail to Gmail is much easier than you think.
Others have said you can setup Gmail so you can "Send As" your old e-mail address and you can set Gmail to download all Hotmail's e-mail.
Also seeI also suggest you read "Comparison with Google Sync" below.
Hotmail supports Exchange ActiveSync for mobiles and DeltaSync for Desktopdesktops. It also supports POP3 for mobilemobiles or desktopdesktops (the only non-propitiatory format it does support).
It won't work in Outlook for Windows, Outlook for Mac or Apple Mail, all of which support Exchange but do not support EAS. If it used to work by entering the Hotmail EAS server details into an Exchange compatible client it's more by luck. It's unlikely to be Microsoft stopping you, the technology isn't designed to work that way.
Microsoft's technical reasons for creating a new technology are sound but lets face it the real reason is to stay propitiatory. If you recall it used to be the case that Hotmail didn't support Exchange ActiveSync or POP3. You could only access Hotmail at hotmail.com.
I doubt they will though, to the best of my knowledge only Hotmail uses it and it's clear Apple would prefer you use MobileMe or Gmail. The blame here rests as much with Apple notas Microsoft.
Apple have added their own Google Sync into MACMac OS X 10.6 Snow Leopard (I think 10.6 not 10.5) and further enhanced it with MACMac OS X 10.8 Lion. They do this using IMAP for Gmail, CalDAV for Google Calendar and I don't know how Google Contacts works but something similar.
Microsoft haven't added zilch Google Sync support to Windows (why would they?). You can use Mozilla Thunderbird (with Lightening calendar add-on) and Mozilla Sunbird though or if you prefer using Outlook there are plenty of third party Google Sync tools.