What Is Yawm al-Hajj al-Akbar?
Question

What is the meaning of the greatest day of Hajj (yawm al-Hajj al-akbar) and the greatest Hajj (al-hajj al-akbar)? Do they mean the same thing? Or are they different from one another? Are both of them to be found in the Quran and Sunnah?
Praise be to Allah.What is Yawm al-Hajj al-akbar? 
The greatest day of Hajj (yawm al-Hajj al-akbar) is the Day of Sacrifice , which is the tenth day of Dhu’l-Hijjah. 
Abu Dawud narrated from Ibn ‘Umar (may Allah be pleased with him) that the Messenger of Allah (peace and blessings of Allah be upon him) stood up on the day of Sacrifice during the Hajj that he performed and said: “What day is this?” They said, “the Day of Sacrifice.” He said, “This is the greatest day of Hajj.” (Sunan Abi Dawud, 1945; classed as sahih by al-Albani in Sahih Abi Dawud, 1700) 
Al-Bukhari (369) narrated that Abu Hurayrah (may Allah be pleased with him) said: Abu Bakr al-Siddiq (may Allah be pleased with him) sent me amongst the announcers to proclaim in Mina on the Day of Sacrifice, “After this year no mushrik is allowed to perform Hajj or to circumambulate the House naked.” 
Why the Day of Sacrifice is called Yawm al-Hajj al-akbar
The Day of Sacrifice is called the greatest day of Hajj because during that night there is the standing at ‘Arafah and staying overnight in al-Mash’ar al-Haram. And during that day there is the stoning of the Jamrah , offering the sacrifice, shaving the head , tawaf and sa’i, which are all actions of Hajj. 
The day of Hajj refers to the time, and the description of greatest refers to the actions done at that time. The greatest day of Hajj is mentioned in the Quran where Allah says (interpretation of the meaning): 
“And a declaration from Allah and His Messenger to mankind on the greatest day (the 10th of Dhul-Hijjah — the 12th month of Islamic calendar).” [al-Tawbah 9:3]
And Allah knows best.