I believe equipment was pretty much the same, except few things:
1 - Operation Overlord/D-day, paratroopers were issued Thompson SMG extensively, but by the time Operation Varsity, M3/M3A1 Grease guns were issued to paratroopers (Thompson was very expensive and complicated to manufacture VS Grease guns were easier and cheaper). I have seen many photos of paratroopers carrying Thompson during D-Day (and I haven't seen pictures of Greasegun being used at this time).
2 - I believe uniforms were also different as well. Paratroopers were issued with M1941, M1942 and M1943 field jackets during WWII. Some were in different shades of Khaki. I just don't know exactly which uniform was worn by which unit and which time frame. But here are some photos for you to review.
This is the famous IKE photo prior to D-Day. These 101st ABN Div paratroopers are wearing M1942 paratrooper uniform, which was standard issue from 1942 until fall of 1944.
There were some issues with M1942 uniform, so riggers of 82nd and 101 ABN came up with Reinforced M1942 uniforms with heavy Olive drab canvas. You can see the patch on elbow on uniform pictured below VS picture with IKE doesn't have the elbow patches. Either one is correct for this time frame (D-Day) since there weren't enough modifications done to M1942 uniform.
photo below is M1943 paratrooper uniform, which was issued from Fall of 1944. Different pocket style and positioning.
I hope this helps! AIRBORNE ALL THE WAY!