There is nothing fancy about an "app" in Salesforce, it is just a collection of tabs with a customised logo.
See help docs https://help.salesforce.com/HTViewHelpDoc?id=dev_tabsetcreate.htm
I have no idea why you would EVER use the New App Wizard - it's just not the way you start planning new functionality in any database, let alone Salesforce.
Have as few apps as possible - maybe one for the organisation with their logo - see New Salesforce Org. I reckon it is better to have an overflow of tabs than to have to switch to a new app.
If you have Salesforce open in multiple tabs, you can't have two separate apps open in separate tabs. As soon as you click on a Salesforce tab, the app will change to the currently selected app - which is why having as few apps as possible is a good thing.
Hide apps from the profile that are really not necessary - it an appexchange app installs an "app" but it's only got one useful tab that you use once in setup, then hide that app, it just takes extra time to look for the app you really need.
Trailhead has a good overview of Tabs https://developer.salesforce.com/trailhead/force_com_introduction/customizing_user_interface/ui_nav_buttons_links