Mistakes were made on both sides, but ultimately, the Germans made more (evidenced by the successful Allied invasion).
1. Elastic vs. Static Defense
Hitler's strategy was to defeat the allied invasion on the beaches. He wanted strong defensive belts (mines, barbed wire, machine gun bunkers, etc) right on the beach. Hitler proposed this type of strategy throughout the war in every theater. Suprisingly, or luckily, it actually worked in a few instances on the Eastern Front; specifically during the Soviet winter offensive of '41 outside Moscow and the defense of Velikye Luke. It would lead to disaster at Stalingrad in '42 but Hitler didn't learn from his mistakes. This type of strategy worked great in WW I, but tactics and weapons had changed in the preceeding 20 years. The problem with that strategy was that it required the Germans to attempt to be strong everywhere, and as the old military maxim goes, "he who tries to hold everything, holds nothing".
Rommel had advocated an elastic defense which would emphasize flexible response. His view was that the beach defenses would be designed to inflict as many casualties as possible and delay the Allies long enough for him to gather the necessary forces in order to counter attack and defeat the invasion. But he was hampered by...
2. Central vs. local command
Hitler wanted personal control of the Armored reserve. He had ordered that they could only be released by his expressed order and he ordered them to be held too far from the coast. When the invasion happened, instead of the Armored forces being available for immediate counter attack, precious hours were wasted waiting on Hitler to give the OK (he was actually taking a nap and had asked not to be wakened), and they had to move a long distance to the staging areas. By the time they arrived, the Allies had secured a foothold.
3. Faulty intelligence
Part of the reason Hitler wanted control of the armored reserve was that the allies tricked him into believing the attack would fall on Calais instead of Normandy. This was due to the poor ability of Germany to establish any type of credible intelligence networks outside of Germany. Some attribute poor german intelligence services to the fact that there was no controlling office, instead, Army intel (The Abwher) often found itself competing against the SD (Gestapo) and SS. These 3 offices were usually at odds instead of coordinating their efforts.
As for the Allies, there mistakes didn't hamper them as much. It is true that the airborne assault was scattered and faulty, but that actually was a benefit as the appearance of paratroopers in so many places confused the Germans as to what their real intention was.
Allied tanks were also outgunned by their german counterparts. But what they lacked in firepower, they made up for in numbers in speed. Although German tanks had larger caliber guns, they were often extremely heavy, slow, and used enormous amounts of precious fuel. Allied tanks, because they didn't possess the amount of armor of the germans, were faster and could disengage from a bad situation and move farther and faster on less fuel. The allies were also able to put more tanks into battle.
The only real tactical allied blunder could be attributed to Montgomery. As was his custom throughout the war, instead of seizing the initiative and exploiting opportunities, he preferred a more patient and set-piece approach which led to the failed assault on Caen against the 12 SS Hitler Jugend Division.