It's not just the actual passport, it's also the additional citizenship that can give other tangible benefits. The biggest benefits off the top of my head:
1) As others mentioned, you can travel visa-free to many countries on one passport but not on another. For example: having a passport from EU country allows you to live,work, or open business in most of the EU countries without a visa. Even American passport holders do not enjoy this privilege in the EU.
2) If you have an emergency situation and one country takes away your passport (for example, Eric Snowden's situation), you can still travel on your other passport.
3) Many countries do not allow extradition of their citizens to some or any other countries. Notable example being that France does not allow extradition of french citizens to the US. This is how Roman Polanski is able to live a playboy lifestyle in France for 3 decades after he has been indicted for some sort of statutory rape crime in California.
There are other benefits as well if you dig deeper.
1) As others mentioned, you can travel visa-free to many countries on one passport but not on another. For example: having a passport from EU country allows you to live,work, or open business in most of the EU countries without a visa. Even American passport holders do not enjoy this privilege in the EU.
2) If you have an emergency situation and one country takes away your passport (for example, Eric Snowden's situation), you can still travel on your other passport.
3) Many countries do not allow extradition of their citizens to some or any other countries. Notable example being that France does not allow extradition of french citizens to the US. This is how Roman Polanski is able to live a playboy lifestyle in France for 3 decades after he has been indicted for some sort of statutory rape crime in California.
There are other benefits as well if you dig deeper.