Is there a browser that allows different instances that don't use the same cookie for the same site?
Solution 1:
IE versions 7 through 10 offer a "New Session" command which may do what you need (but it depends partly on how the site sets/uses its cookies). You can also set your IE shortcuts to automatically open in a new session by adding -nomerge to the end of the Target line.
Most current browsers also offer a "private" mode (IE Tools|InPrivate Browsing, Firefox Tools|Start Private Browsing, Chrome [wrench]|New incognito window, ...).
And as others have mentioned, in Firefox you can use profiles to achieve what you want.
Solution 2:
Firefox has an extension that could do that:
Multifox is an extension that allows Firefox to connect to websites using different user names. Simultaneously!
For example, if you have multiple Gmail accounts, you can open them all at the same time. Each Firefox window, managed by Multifox, accesses an account without interfering each other
Each Multifox window is flagged with a number indicating the identity profile. Logins made in windows with different numbers are isolated.
Logins are preserved the same way they are in “regular” windows. Even if you close the window or quit Firefox.
The identity profile of each window is preserved when Firefox restores the session.
It can also easily switch between profiles:
Solution 3:
If you are using Firefox (I assume you are since you mention it) you could just set up different profiles. Each profile has it's own set of cookies, plugins, configuration, etc.
I'm not sure you can run two instances at the same time on different profiles, but you haven't made it clear if that is also required.
First you need to access the Profile Manager by running firefox.exe -profilemanager, from here you can create any number of new profiles and rename those you already have.
Then you should untick the Don't ask at startup box, then the Profile Manager will appear every time you run Firefox, allowing you to select the profile you need everytime it starts.