Firstly: 

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 Allaah be upon him) said: “Whoever prays Fajr in congregation then sits remembering Allaah 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). 

There is some difference of opinion as to the soundness of this hadeeth. A number of scholars classed it as weak, and others classed it as hasan. Among those who classed it as hasan was al-Albaani (may Allaah have mercy on him) in Saheeh Sunan al-Tirmidhi.  

Shaykh Ibn Baaz (may Allaah have mercy on him) was asked about it and he said: This hadeeth has some isnaads 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.

End quote from Fataawa al-Shaykh Ibn Baaz (25/171). 

The apparent meaning of this hadeeth is that the one who does that will have the reward of a complete Hajj and ‘umrah, and this is a bounty that Allaah bestows upon whomsoever He wills. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked: It says in the hadeeth, “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 hadeeth is subject to further discussion. Many scholars classed it as daeef (weak). Secondly: if we assume that it is saheeh, the reward of Allaah is not subject to analogy. A person may be rewarded greatly for a small deed, because the reward is a bounty from Allaah that He gives to whomsoever He will.

End quote from al-Liqa’ al-Shahri (74/22).  

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 Allaah and praying is likened to Hajj in terms of reward only. The hadeeth 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 Laa ilaaha ill-Allaah wahdahu laa shareeka lah (there is no god but Allaah alone with no partner or associate) one hundred times – it is like freeing ten slaves. But if a person was obliged to offer kafaarat yameen (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 hadeeth is aimed at encouraging us to remember Allaah, may He be exalted, and to sit in the mosque until sunrise, and do two rak’ahs after that.