Firstly some solutions (some named previously) -
Switch to Gmail: This is your best long term solution. Mac OS X 10.6 Snow Leopard and Mac 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 10.6 Snow Leopard and comments above says 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. Hotmail 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.
This is a pain though (especially if you've got subfolders). If you can get hold of a Windows PC temporarily there is a MUCH easier way. Open Outlook or Windows Live Mail and add Hotmail, also add Gmail using IMAP.
It's then a simple case of dragging and dropping the Inbox and Sent Items contents from one to the other, then dragging each subfolder from one to the other.
This also works when migrating from any IMAP compatible e-mail provider to another IMAP compatible e-mail provider.
Finally Google has equivalent filtering rules, you'll only need to set them up once.
I also suggest you read "Comparison with Google Sync" below.
Now to debunk some inaccuracies stated above :-) -
Hotmail supports Exchange ActiveSync for mobiles and DeltaSync for desktops. It also supports POP3 for mobiles or desktops (the only non-propitiatory format it does support).
For all those people who think Exchange support also means Exchange ActiveSync. It doesn't! EAS is an extension of Exchange, it is not Exchange!
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.
You'd think they could make EAS work on desktops too but literally no desktop client has ever supported EAS (Mac or Windows), there's got to be some reason why that's the case. It could be a technical one or it could be Microsoft doesn't want you using a mobile technology on desktops. We'll never know.
Microsoft are on record as saying IMAP is an old technology and DeltaSync is much better. This is true. It's able to synchronise e-mail, contacts, calendar, tasks and I think notes. It's also much more efficient than IMAP at handling e-mail.
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.
Microsoft aren't going to support IMAP, it's not going to happen.
The trouble is Exchange, Exchange ActiveSync and DeltaSync are propitiatory which require a licence. Apple licensed Exchange and Exchange ActiveSync from Microsoft. There's no reason why Apple couldn't licence DeltaSync (that I know of).
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 as Microsoft.
Exchange brings in potential big business support and EAS is the universal mobile standard for e-mail/calendar/contacts sync. DeltaSync just gives Apple Hotmail.
Comparison with Google Sync -
Google Sync supports EAS, POP3, IMAP and CalDAV. Using the latter two methods you can get proper e-mail sync and calendar sync. Outlook or Windows Live Hotmail don't support CalDAV but Apple iCal, Mozilla Thunderbird (with Lightening calendar add-on) and Mozilla Sunbird do.
To my knowledge there's no way to get contact sync except the business Google Apps.
Apple have added their own Google Sync into Mac OS X 10.6 Snow Leopard (I think 10.6 not 10.5) and further enhanced it with Mac 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.
I use GSyncit to synchronise Google Calendar/Google Contacts which works brilliantly and obviously Outlook natively supports IMAP.
The best you can expect with Google Sync and Windows Live Mail is Gmail IMAP, no Google Calendar Sync or Google Contacts Sync.