Windows Live Mail is an email client used to access email accounts. Windows Live Hotmail is an email host/provider, being ported to Outlook.com Cloud Services.
http://answers.microsoft.com/en-us/...and-live/8d10dfe9-d61a-44a1-b18d-18e714a5719f - note the clarification by an MS Mod.
MS is porting the entire Outlook.com Cloud Services backend to the same system as Office 365. I use both Outlook.com (and can still login with my Live username) and Office 365.
Regarding moving to the Mac, there are issues with synching mail and contacts to the Mac apps, and they're long-standing issues going back several years. MS has promised - in their roadmaps - fixes to these synching issues, but I'm not going to hold my breath. A significant issue with Mail is delayed synchronization - Outlook.com's recent move to fully EWS will likely further complicate email synching between MS's products and non-MS clients; I find this a bit aggravating as I can make significant backend settings changes to my iCloud accounts directly from Outlook 2016 for Windows and the installed (Windows 10) Mail app. The Mac Contacts app has long had sync/timeout issues (largely related to resolving Categories), pretty much going back to Day 1 - with Exchange Servers (I own a Server 2012 license), Office 365, and Outlook.com, and only MS can fix this.
My solution is to use Outlook 2016. It works. I've learned to use Outlook with MS products, Airmail or the Gmail web interface with Gmail, and the Apple clients with Apple services - each provider uses different data fields, and each provider keeps changing their backend. I'm done fighting that battle, and I've had zero issues since I switched to using this workflow.
An alternative? Sign up for a free month of Office 365, download and install a copy of Outlook 2016 and sync it to your MS account. Wait for a bit. Drag all of your contact/calendar data to a place on your Mac. Set up an iCloud account, then import the contacts and calendars into your new iCloud account, then set up Mail to to talk to your MS account. I find this to be too much of a PITA - Outlook.com/Live Mail uses a completely different data format than what Apple defaults to, and I can pretty much guarantee your data will get bunged up sometime in the future. Outlook (the desktop app) fixes that headache for a lot less coin than a couple of hours of therapy... Cheers!