The Iron Cross
The Iron Cross is not only older than the NSDAP, but older than Germany itself. It is still in use as the symbol of the German Bundeswehr, which would be quite out of the question had it been a Nazi symbol.
In 1813, Friedrich Wilhelm III of Prussia established the decoration Das Eiserne Kreuz, to be awarded to soldiers during the war against Napoleon. The cross itself came from the medieval Teutonic Order, which had used variations of the Mantuan cross, not to be confused with the Maltese cross. The Iron Cross came into use again during the war with France 1870-71, during the First World War (when it was awarded to Hitler, Göring and a few million other German soldiers) and again during the Second World War.
It was awarded to soldiers during war - and will probably be again, should Germany find itself in another armed conflict. It was certainly no bling-bling to be worn by your average Nazi, not even in the party elite. Hitler was very proud over his crosses, and would never have allowed it to be awarded on political merits - and why should he, when it wasn't a political symbol? It was carried by many officers who were anything but national socialists (on aristocratic grounds though, rather than democratic). It was awarded to a number of Jews, and in 1914 Hugo Gutmann, Jew and immediate superior officer of Adolf Hitler, recommended the cross to be awarded to him.
In 1945, Nazi symbols were outlawed in Germany. One exception were Iron Crosses from WWII, but not because the crosses themselves were Nazi symbols, but because the 1939 version had been decorated with a small swastika. (Note: Hitler & co thus had no swastikas in their crosses.)
In 1957 these Iron Crosses were outlawed as well, or rather, the exception for the little swastika was removed. Instead, they were exchanged for Iron Crosses with oak leafs (which had been used on all Iron Crosses prior to the 1939 model). The year "1939" remained in place, though. These crosses are perfectly legal to wear since they aren't Nazi symbols.
Another de-Nazified decoration was the German Cross in Gold. The 1957 version to the right.
Had the Iron Cross been a Nazi symbol, the German Bundeswehr would have an illegal logotype.
It can be argued that the Iron Cross wasn't Nazi to start with, but was turned into a Nazi symbol in the same way as the ancient swastika. But unlike the swastika and the eagle, the Nazis never used the Iron Cross as a symbol for the party, national socialism, or anything such. When the cross did symbolise something, i.e. apart from its use as a military decoration, it was the German armed forces, neither more nor less. If post-war Germany can keep track of the difference, we should too.
That some less well informed racists have mixed up these facts as well shouldn't matter. (One might, though, ponder how Hitler would react, had he seen kids walk around with the military medal of bravery he risked his life for...)
The Maltese Cross
This is a completely different cross, with no relation whatsoever to the Iron Cross. The Maltese Cross has a rich history indeed, is used in many countries and on Malta in particular. Above, the Swedish Order of the North Star. Below, some more.
Thanks to many helpful people on Skalman