Given the single market and many of the EU directives that apply are there any benefits other then patriotism or nationalism that would advocate for having two or more EU citizenships?
Inside the EU, they should have the same benefits, so if you don't plan on leaving the EU, it should not matter.
However countires outside of the EU doesn't look at the EU as a single country, so they can have different (reciprocal) agreements. This includes getting tourist and also work visas outside of the EU. In this case you might get some benefit of having multiple citizenships.
For example if you get a British cizienship you also become a citizen in one of the Commonwealth countries, making it easier to travel to other Commonwealth nations.
An other example is the Visa Waiver Program in the USA, which means that for some countries you don't need to get a visa to travel to the USA for short term. Poland is still not inside this program, but most other EU countries are, so if you only have a Polish Citizenship you cannot benefit from it.
Also note that in some EU countries euroskepticism is becomming prevalent - "Brexit" is already a reality - meaning it might be possible, that an EU country will leave the EU. Although this hasn't happened before(*) so we don't know how the laws will change, but probably this would also mean that if you only have a citizenship in a country that left the EU, you would probably lose the EU citizenship as well.
(*) Except for Greenland leaving the EEC in 1982 as noted by @gerrit. Note that although Brexit is in progress, we still don't know how it will end, and what will happen afterwards.
To add on to StzupY's answer, a country's nationals often have a much easier time getting things done. That means, even if the law eventually treats everyone practically the same, foreign EU-nationals usually have to deal with more paperwork and longer processing times.
I also think that there are still enough differences in civil law, that could have a feelable impact on your life. For example, Polish naming law allows for husband and wife to have two different last-names, taking into account the masculin and feminin suffix. German naming law requires husband and wife to have identical last names. So for example in Polish, I am Cichocki, and my wife would be Cichocka. If we married under German name-law, my wife would be called Cichocki - which in Polish sounds ridiculous.
I've just started this family-thingy ; ), but I'm prepared to deal with lots of more little nuances, because my wife and I are of different EU-nationalities.
As the others have said, the differences are mainly going to be perceptible outside the EU. But even within the union, some countries restrict some of their benefits to actual nationals: for instance, I'm a French citizen living in Sweden, and I can't benefit from their study grant program. I'd have to have the Swedish nationality.
This is the only example I can think of, but I'm sure there are a couple other instances where being an actual national grant you more rights than just being a member state citizen.
One advantage could be the work permition - there were some restrictions for citizens of Bulgaria and Romania until 2014, and there are still restrictions for citizens of Croatia - only 8 EU countries don't restrict Croatian workers. So Croatian citizens (and most likely citizens of upcoming EU members) will benefit from dual EU citizenship
Another point are the elections - you cannot vote on all elections in the country where you're only resident but not a citizen.
When you have nationality (and citizenship) of one of the European country, you are automatically granted also the European Citizenship.
But the European Citizenship do not make you equal with the Citizens of another EU country.
The rights of a European citizen in european country where he is not citizen can be checked here: http://en.wikipedia.org/wiki/Citizenship_of_the_European_Union
So the more states you are a citizen the more advantages you get in those states.