It depends on what you're using the cable for. If you're using it for any flavor of Ethernet (10/100/1000BASE-T), then Yes, it makes a difference for signal integrity.
Ethernet uses differential signaling, where a signal waveform is sent down one wire of each pair, and the inverse of that signal is sent down the other wire of that pair. Because the two wires of the pair are kept so close together by the twisting, each wire is equally exposed to any interference. At the receiver, the inverse signal (with interference) is inverted again and combined with the normal signal (with interference) from the other wire. This causes the interference to be canceled out, while the signal is reinforced.
If you don't get the twisted pairings right, the wires will be exposed to different levels of interference, so the cancellation won't work as well. This can make a difference for longer runs through more interference-prone environments (past lots of other cables and electrical/electronic equipment), and at higher speeds.
People ask questions on here all the time about why their long gigabit Ethernet run isn't working, and the solution is often that they got the pinout right, but didn't get the twisted pairings right.
So yes, make sure pins 3&6 are a twisted pair, etc.
Nice and clear explanation... It was the 3 and 6 separation that made me wonder in the first place... I am running short runs of 20 meters (at most) in the same room with 10 cables (at most), running through one pipe. That is the reason it was working without the 3&6 (green) from one twisted pair. – Sunny – 2015-05-01T10:57:11.547
1For gigabit ethernet all 4 pairs are needed. For lower speeds sometimes 4&5 were used for POTS telephony, so you could stick an RJ11 into the RJ45 socket for your phone. – wurtel – 2015-05-01T11:08:26.957