It’s not such a straight forward question to answer as many of the answers on posted here would have you believe. The units themselves are actually tasked with different operations; however a brief lesson about the 2 units is in order.
History
Firstly the British Special Air Service Regiments (SAS)
In 1941 Lieutenant David Stirling and Lieutenant Jock Lewis gathered a small unit of 66 volunteers. This unit was named the L. Detachment of the Special Air Service Brigade. The main objective of this group was to conduct raids against the lines of communication of Axis forces in the Western Dessert. T
The 2nd SAS Regiment was raised and commanded by William Stirling, David's brother. The 2nd regiment was responsible for the Operation Torch landings in Algeria.
Since then the SAS has grown and shrunk over the years to its current size.
Secondly the 1st Special Forces Operational Detachment-Delta (1st SFOD-D) — commonly known as Delta in the US Army, Delta Force by civilians, and Combat Applications Group by the US Department of Defence.
The unit was started by Colonel Charles Beckwith in 1977. Throughout its creation, the unit had the benefit of experience from the British SAS.
The unit had no real combat experience until Operation Eagle Claw the failed attempt to rescue American hostages from the U.S. Embassy in Tehran in 1980. The mission failed due to an overly complex plan, inadequate Special Operations Aviation training for the accompanying aircrews, a collision between a rescue helicopter and a refuelling tanker aircraft, and mechanical problems with the helicopters transporting the rescuers, caused by a severe sandstorm which clogged the engine intakes and reduced the pilots' visibility to nearly nothing.
Organization
The SAS is made up of 3 regiments and several independent squadrons;
22 SAS is the primary unit and includes HQ Squadron, 5 Sabre Squadrons A, B, D and G Squadrons supplying the bulk of 22 SAS troops. R Squadron, which is composed of ex-SAS Troopers and is for casualty replacement, and 1 Pagoda Squadron responsible for Counter Revolutionary Warfare and anti-terrorist duties.
21 SAS is made up of former soldiers and includes HQ Squadron and 3 Sabre Squadrons A, C and E
23 SAS is the units TA formation and includes and HQ Squadron and 3 Sabre Squadrons A, B and C
Delta Force is made up of 4 Squadrons;
A, B and C Squadrons are Deltas “Operational” units, split 70/30 between Assault teams and recon teams.
D Squadron contains about 70% of Deltas personnel and is tasked with all support roles, Mechanics, communications, intelligence analyse.
Recruitment
22 SAS recruits only serving soldiers with more than 3 years service under their belt, the vast majority of recruits come from elite British army units such as the Parachute Regiment.
21 SAS recruits only former serving members of 22 SAS
23 SAS recruits civilians but has a high percentage of ex –military
Delta recruits only serving soldiers with 2 ½ years service under their belt (although they may come from the National Guard or Reserves) However a high proportion (around 80%) come from units such as the Green Berets and the 75th Ranger Regiment.
Roles
The SAS is primarily tasked with Long Range and Deep strike operations, Long range Reconnaissance and has a limited CRW and anti-terrorist functionality.
Delta is primarily tasked with CRW and anti-terrorism with a limited Deep strike and Reconnaissance role.
Both units train together in all roles, and both units practice “Hearts & Minds”, including the training of friendly nations Special Forces and Paramilitary Groups.
Operational History
Although combat operations of both units are kept tightly under-wraps it’s know that the SAS is far more active than Delta. Primarily the British rely heavily on Special Operations Forces whereas the Americans have always been more reluctant to field SF units. When it comes to well known operations Delta have had far worse luck than their British counterparts having been involved in several well publicised operations that have ended badly for the force.
Overall the SAS would be the better unit, they have taken part (and continue to do so) in more Combat operations than Delta, and real operations are ALWAYS better than training. They have had a higher success rate than Delta, giving the men of the SAS higher morale. However both units continue to share men between them on exchange programs, and both units train and operate together often.
So the SAS might have the edge….but the UK and US are so close operationally that it really doesn’t matter that much.