What makes the Marine Corps 'special' is a bit of a complex term - and in a lot of ways, it's more tied to a sociological study as many factors from effective advertisement to the Marine Corps differentiating themselves from other branches in various ways that all add up to why people may consider the USMC 'special.'

Historically speaking, the Marines - who trace their heritage officially back to November 10th, 1775 when the Second Continental Congress - were designed to operate as naval infantry.

Though small, with the Marines opening 1800 with just 368 Marines, they were an asset to the fledgling country: as professional soldiers onboard naval warships, they were given tasks that the army simply wasn't around for or capable of.

For instance, in the Barbary Wars, the Marines successfully stormed the city of Derna in Tripolitana (modern day Libya). From the link above:

Lieutenant Presley O'Bannon and his Marines marched across 600 miles of the Libyan Desert to successfully storm the fortified Tripolitan city of Derna and rescue the kidnapped crew of the USS Philadelphia. The Marines' victory helped Prince Hamet Bey reclaim his rightful throne as ruler of Tripoli. In gratitude, he presented his Mameluke sword to Lt O'Bannon.

This famous sword became part of the officer uniform in 1825, and remains the oldest ceremonial weapon in use by United States forces today.

The Battle of Derna was the Marines' first land battle on foreign soil and is notably recalled in the first verse of the Marines' Hymn: "From the Halls of Montezuma to the shores of Tripoli, we fight our country's battles in the air, on land and sea."

You'll note that a lot of emphasis is put on the history and lore of the Marines from other Marines - a big differentiating mark from other branches. The mameluke sword, the Marine Corps Hymn, and so on are all traced back to various events in Marine Corps history.

It's also worth noting that the Marine Corps has always been the smallest of the branches (minus the Coast Guard). For instance, in 1945, at the end of WW2, the Army had 8.3 million men in uniform, the Navy had 3.4 million... while the Marines had 475,000.

Despite those small numbers, the Marines had fought many fierce battles in the Pacific which bolstered their reputation. Not to mention, in the Pacific, the Marines were involved in major offensive operations against the Japanese before the Army got involved in major operations in Europe. And when the war in Europe was drawing to a close, one of - if not the most iconic image of the war - was of the Marines with Raising the Flag on Iwo Jima.

It's worth noting too that, in the post WW2 the Marines are the only branch that operate on air and land and sea in a combined force operations (minus the Marines operating from Navy ships.. hence the armed forces joke that Marines stands for "My Ass Rides in Navy Equipment, Sir") - something that differentiates the other branches. Sure, the Army has helicopters too - but it doesn't have the fixed wing strike aircraft that the Marines do.

You may argue that them having higher PT standards today, their officers going through far more training than other branches (e.g., all officers are required to attend the 6 month course known as The Basic School), longer boot camp than others, etc. all contribute to this differentiation as well today.

As I said, all of this is hard to quantify into what makes the USMC considered 'special' but as I've mentioned above, the image of 'elite' combined with a history of small numbers engaged in fierce battles combined with unique traits that differentiate it from the other branches are all factors.

Part of the modern reputation of the USMC owes a great deal to The Battle of Belleau Wood.

The American Expeditionary Forces under Gen. John J Pershing arrived in Europe in 1917. Pershing's censorship policy required that no specific unit be identified in reporting and Army censors were adept at their job. The result for readers back home was a near complete lack of detail about the progress of our boys over there (https://youtu.be/B6hRDS3LvQQ).

Enter Floyd Gibbons, a well known war correspondent in WWI the way Ernie Pyle would be later in WW2.

As American forces (Army and Marine) engaged the Germans in Belleau Wood, Gibbons advanced to the front and was seriously wounded. His dispatches were sent to the rear along with word that he had been killed by German machine gun fire. Although this turned out to be exaggerated (he lost an eye, sustained a skull fracture, and was shot in the shoulder and arm) by the time his articles reached the censors they allowed them to go through, reportedly as a mark of respect for the supposedly departed correspondent. For three days dispatches flowed from the front, detailing the exploits of the Marines before censorship was reapplied.

By that time, Gibbons' uncensored dispatches were on the way home and went to print with the details of the Marine units in action. Doing so put a name and face on the American war effort and placed the Marines front and center.

http://www.worldwar1.com/sffgbw.htm , condensing the 1953 biography "Floyd Gibbons - Your Headline Hunter," (https://www.amazon.com/Floyd-Gibbons-Your-Headline-Hunter/dp/1258173182).

(Sidenote: After the war's end, Pershing commissioned a statue from a French sculptor who used a Marine as his model. When the statue was presented, Pershing refused to purchase it since the Marine emblem had been included on the helmet. Marine General Smedley Butler organized a fundraiser and purchased the piece, which still stands outside the Marine Headquarters building in Quantico, Virginia.)