Reward of Hajj and ‘Umrah after Fajr
Question

What is the difference between Hajj and sitting after Fajr prayer until the sun rises, then praying two rak’ahs, as the Messenger (peace and blessings of Allah be upon him) told us that the one who does that will have a reward like that of a complete, complete, complete Hajj?
Praise be to Allah.Reward for staying up after Fajr
With regard to sitting after Fajr prayer until the sun rises then praying two rak’ahs, it is narrated that the Prophet (peace and blessings of Allah be upon him) said: “Whoever prays Fajr in congregation then sits remembering Allah until the sun rises, then prays two rak’ahs, will have a reward like that of Hajj and ‘Umrah, complete, complete, complete.” (Narrated by al-Tirmidhi, 586) 
Degree of the hadith about reward for staying up after Fajr
There is some difference of opinion as to the soundness of this hadith. A number of scholars classed it as weak, and others classed it as hasan. Among those who classed it as hasan was al-Albani (may Allah have mercy on him) in Sahih Sunan al-Tirmidhi.  
Shaykh Ibn Baz (may Allah have mercy on him) was asked about it and he said: 
“This hadith has some isnads with which there is nothing wrong, thus it is regarded as hasan li ghayrihi (hasan because of corroborating evidence), and it is mustahabb to offer this prayer after the sun has risen to the height of a spear, i.e., approximately fifteen or twenty minutes after sunrise.” (Fatawa al-Shaykh Ibn Baz, 25/171)
Meaning of the hadith about reward for staying up after Fajr
The apparent meaning of this hadith is that the one who does that will have the reward of a complete Hajj and ‘umrah, and this is a bounty that Allah bestows upon whomsoever He wills. 
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) was asked: It says in the hadith, “The one who sits in the place where he prayed after Fajr prayer until the sun rises, it will be like a complete, complete Hajj and ‘umrah,” or words to that effect. Does this mean that the one who does that will have the reward like that of Hajj and ‘umrah, or what does it mean? 
He replied:  
“Firstly: this hadith is subject to further discussion. Many scholars classed it as da’if (weak). Secondly: if we assume that it is sahih, the reward of Allah is not subject to analogy. A person may be rewarded greatly for a small deed, because the reward is a bounty from Allah that He gives to whomsoever He wills.”  (Al-Liqa al-Shahri, 74/22)
Difference between sitting after Fajr till sunrise and performing Hajj and ‘Umrah
As for the difference between this sitting and performing Hajj and ‘Umrah, Hajj involves spending money, striving physically and putting up with hardships, and it is obligatory for the one who is able to do it, and it is one of the pillars of Islam. This sitting, remembering Allah and praying is likened to Hajj in terms of reward only. The hadith does not mean that the one who does that has performed Hajj and ‘umrah and is no longer obliged to do them. 
This is similar to the case if one says La ilaha ill-Allah wahdahu la sharika lah (there is no god but Allah alone with no partner or associate) one hundred times – it is like freeing ten slaves. But if a person was obliged to offer kaffarat yamin (expiation for breaking an oath, one form of which is freeing a slave), and he recited this dhikr, that would not count instead of freeing the slave as required. 
It is well known that the scholars said concerning such cases: The likeness is in reward, not in discharge of one’s duty. 
The meaning of this hadith is aimed at encouraging us to remember Allah, may He be exalted, and to sit in the mosque until sunrise, and do two rak’ahs after that.
And Allah knows best.