You can open multiple different projects with Unity at the same time; however, opening the same project twice could ruin the project in previous versions and so I guess they've added a check to prevent this (seems like that check only is implemented for Windows because I can't reproduce this here on my Mac).
If you want to open "the same" project twice, you can make a copy of the project, which can be very useful for debugging networked games on a single machine. If you're using Asset Server or another version control system, it may be a good idea to keep both versions under version control - that way, you can make changes on both "sides" and still keep everything in sync pretty conveniently.
In the Unity Preferences, there's a checkbox "Show Project Wizard at Startup" - if that's checked, Unity opens that wizard instead of opening the previously opened project; so that should solve the issue for you. On Mac OS X, you can also cmd+click (press the cmd/Apple key and click) the app-icon to achieve the same effect without changing this setting. Might work under Windows as well (probably would be ctrl+click but if that doesn't work, I'd try shift+click and alt+click as well) - but I can't try that because I don't have Unity on a Windows machine ATM, so I'm not sure.
On Mac OS X, when you double click the app-icon while the application is already running, you simply "focus" the opened instance. So you need a little trick to open two instances of Unity (Windows handles this differently, so it's easier to open multiple instances there - just double click twice and you got two instances running):
Either right click on the app-icon in the Finder, and select "Show Package contents", the browse to "Contents/MacOS" and double click "Unity" (which has a console Icon there). That does in fact open a console and you got your second instance. Which kind of leads to the second way of doing this:
You can also open Unity directly from the console and this also allows you to give the project path; so you could create a couple of scripts for directly opening a couple of different projects (that's what I did for working on my client-server based game):
#!/bin/bash
/Applications/Unity/Unity.app/Contents/MacOS/Unity -projectPath "/path/to/your/projectA"
If you put that into a file (e.g. startProjectA.sh) and make it executable (chmod a+x startProjectA.sh), you're all set ;-) ... ah, simplest way to create such a script (IMHO): in the console, type vim startProjectA.sh, then type "i" (to get into insert mode), type the script, when you're done, hit "esc", then :wq (: lets you type commands, w is write, q is quit). I think vim is pretty cool but it takes a moment to get used to it.
