I hold dual citizenship (Canada and UK) and do travel on two passports. My principal passport of preference is my Canadian one but there are instances where the UK/European Union is essential or more efficient.
The UK passport allows me to bypass some of the long line ups for foreign nationals when arriving at UK and it allows multiple and cheaper entries (don't ask me why) into some countries (e.g. Qatar) that my Canadian passport does not.
On the other hand, getting back into Canada with just a UK passport does cause problems at times (20 questions).
My parents sometimes used two passports so that their travels to countries like Israel would not jeopardize their safety or entry to countries that have a dislike for those who visit Israel. (use one passport for one set of countries and another for the other set). I have heard that some countries will issue two passports just to allow this sort of scenario.
I am a legal citizen of both countries so enjoy the benefits of citizenship in both.