I think the crux of what you want to know is:
When should one use Outlook, and when Hotmail?
If your primary concern is getting your web
outlook.com email account synced, then you can choose either. Hotmail.com has been upgraded and re-branded to Outlook.com.
However, the brand name
Outlook is more than just the web client
From the Wikipedia page about Outlook:
It can be used as a stand-alone application, or can work with Microsoft Exchange Server and Microsoft SharePoint Server for multiple users in an organization, such as shared mailboxes and calendars, Exchange public folders, SharePoint lists, and meeting schedules. There are third-party add-on applications that integrate Outlook with devices such as BlackBerry mobile phones and with other software such as Office and Skype internet communication.
In other words, if you need to sync with Office 365 or MS exchange, you should pick Outlook as the account to add.