What is the significance of a cross with two cross-bars to Christians? Does it carry different symbolism than a normal cross?
The "double" cross is known as a Patriarchal cross and is well described in the Wikipedia article. There's no point in reproducing more than a sample here:
The Patriarchal cross is a variant of the Christian cross, the religious symbol of Christianity. Similar to the familiar Latin cross, the Patriarchal cross possesses a smaller crossbar placed above the main one, so that both crossbars are near the top. Sometimes the patriarchal cross has a short, slanted crosspiece near its foot (Orthodox cross). This slanted, lower crosspiece often appears in Byzantine Greek and Eastern European iconography, as well as Eastern Orthodox churches.
The top beam represents the plaque bearing the inscription "Jesus of Nazareth, King of the Jews" (often abbreviated in the Latinate "INRI", and in the Greek as "INBI").
Incidentally, the figure in the second photograph is Saint Stephen, first King of Hungary: this is his statue in Budapest. He has a halo and is wearing St Stephen's Crown, characterised by its crooked cross. This in itself answers the question!