Its importance is much more than symbolical; Donald Sensing thinks that Russia would then win the war and dominate all of Europe:
The Soviets certainly would not have slacked their offensives had Normandy failed. If anything, they would have pressed all the harder, but would have pressed equally hard for a much larger share of American war production, insisting that they were making better use of it than we were. As they would have been the only dog in the fight, the demands would have been hard for Roosevelt to resist. Not only would all Germany have become communist, so would France, whose communist cells were very active and which would have benefitted greatly from having the Soviet army literally next door. Imagine the Iron Curtain falling at the English Channel. The Soviet bear would have easily swallowed countries like Denmark, The Netherlands and Belgium. Likewise, Greece's postwar communist insurgency would have succeeded. Italy might easily have turned communist also.Whether you think that this analysis is correct or not, there is little doubt that a defeat of the Allies in Normandy would have meant a very different course of history.
So yes, the world in which we live today is in great part due to those men who fought on the beaches and flatlands of Normandy during a few days of June 1944.