The most important difference between 1944 and today would be in the realm of guided munitions. I once heard that a single F-15 packs as much firepower as an entire squadron of WWII era bombers, when you take into account explosive weight and the percentage of ordnance you can get on target (Keep in mind, the F-15 is a Fighter/Bomber, not a dedicated bomber. If we start talking about the B-52, things get even crazier). Additionally, Naval Gun Fire support has come a long way since the 1940's. US destroyers and cruisers now only come equipped with one or two 5" main guns. In the 1940's, 5" guns were almost considered an afterthought. With improved fuses and nearly automatic rates of fire that can be achieved with today's weapons, you wouldn't need the hours and hours of shelling they used during WW2 landings.
As far as the landings go, with today's amphibious landing tactics and equipment, you wouldn't NEED to land at Omaha beach at all.
This is an LCAC (Landing Craft Air Cushioned). It is just one of the many ways the US Navy and US Marine Corps get troops from ship to shore. The main difference between an LCAC and the landing craft of yore is the fact that the LCAC can access almost any beach in the world, and can travel across dry land. Furthermore, it can achieve incredible rates of speed compared to the Amtracks of WW2 (I think around 70 knots when not weighed down much). Today the US would be able to basically avoid any defensive strongpoints and just stick their landing forces where ever they figured was the least defended.
Helicopters, in widespread use since the Vietnam War, allow entire infantry companies and battalions to be shuffled about at incredible speed compared to the 1940s.
The M1A2 Abrams Main Battle Tank would probably be as close to invulnerable as anything ever employed in warfare. The only reasonable option for destroying one with 1944 equipment would be swarming it with infantry and trying to get a grenade inside. This technique was costly during WW2. Against an Abrams, with a wingman that can just shower his buddy with HE rounds that do nothing substantial to the armor...
As far as the individual soldier is concerned, the primary difference is the body armor. Ceramic plates and flak jackets have greatly increased the survivability of the infantryman. Back in WW2, your armor was a millimeter of cloth. Today it contains plates that would actually be capable of stopping pretty much any small arms round the Wehrmacht utilized (7.62 AP is the limit, I believe). A quick look at the WW2 Killed/Wounded ratio [1:1.65] versus the Operation Iraqi Freedom Killed/Wounded ratio [1:7.3] shows that even if nothing but the current body armor was added to the equation, it is likely that the US would have reduced the number of soldiers killed on D-Day from 2,500 to probably around 700. On the flip side, the infantry of WW2 would be much faster and more agile, as they weren't towing around 50+ lbs of gear. So you have a classic heavy infantry vs light infantry situation here.
The Mk19 Automatic Grenade launcher. Designed for use against troops in the open, troops in trench-lines, light armored vehicles, urban strongpoints, and light fortifications, this 76.2 lb beast is technically man-portable (by someone's standard) and is widely employed on mounted assets. Capable of firing 325-375 40mm grenades per minute, there is arguably no more intimidating weapon in the US arsenal that is commonly used in firefights. I have personally been within 25 meters or so of the beaten zone of someone unleashing a long burst of grenades, and it was, shall we say, disconcerting. This is probably the one weapon capable of allowing an individual to singlehandedly end a firefight.
Today many infantry companies will have communication assets down to the fire team level. This allows for much faster response times to dealing with threats or re-organizing after a firefight or simply getting troops to move around where you want them (radios at the platoon level were very rare during World War 2, and what was in play was of limited range and had no encryption capabilities. When I was in a motorized heavy weapons platoon, we had a dozen PRC-119's, satcom radios, Blue Force Trackers, etc; we probably had comm capabilities that entire divisions during WW2 would have drooled over. And we had 40 dudes).
While the small arms themselves haven't really come a long way, the accoutrements certainly have. Every infantryman today is probably equipped with, at minimum, a 4x scope, NVG's, and a laser for use with night vision. One out of every 4 infantrymen will have a grenade launcher. Another one will have a light machine gun. This allows for the ability to achieve combined arms effects using just a single fire team. And the night-fighting capability, with nothing else, would be a game changer.
The one thing we would be at a disadvantage in would be combat experience. The Germans had been fighting for FIVE years by the time the US actually got into France. Of course, this was an issue during the actual D-Day landings, and didn't hamper things too much, probably because the allies were facing off against the JV squad, so to speak. At the same time, our military back then was well trained for large scale battles, as opposed to how the US military is organized today. Whether or not the current infantryman would fare well is anyone's guess.
Free Fun Fact:
One thing that hasn't changed is the M2 .50 Caliber Heavy Machine Gun. Supposedly something like 95% of the M2s in use currently were originally built during World War 2. The ammunition, however, has received quite the upgrade (SLAP, API, Raufuss, all fun stuff)
Another Fun Fact:
The United States uses a military doctrine termed "Rapid Domination" (Shock and Awe for the soundbite term). The Gulf War and the initial invasion of Iraq during OIF are two examples of this doctrine in use. The basic concept involves gaining air superiority, using tactical and strategic bombers to disrupt and destroy enemy command and control, employing a wide range of offensive maneuvers (amphibious landings, paratrooper drops, armored thrusts, infantry assaults on defensive positions) simultaneously in order to paralyze any decision making ability of the opponent. This military doctrine is heavily based on the so-called Blitzkrieg doctrine of Nazi Germany.