What Is the Difference Between al-Aqsa and the Dome of the Rock?
Question

I have recently received an Email advising me of the status of al-Masjid alAqsa and differentiating it from the Dome of the Rock.
Can you please clarify the situation and advise, if al-Masjid  al-Aqsa is different from the Dome of the Rock, why do we see its picture representing al-Masjid al-Aqsa at all Islamic places, and I (and many other Muslims) were completely unaware of the difference?
Praise be to Allah.What is Al-Masjid al-Aqsa?
Al-Masjid al-Aqsa (in Jerusalem) was the first of the two qiblahs (prayer direction), and is one of the three mosques to which people may travel for the purpose of worship. 
Who built al-Masjid al-Aqsa?
And it was said that Al-Aqsa Mosque was built by Sulayman (peace be upon him), as stated in Sunan al-Nasai (693) and classed as sahih by al-Albani in Sahih al-Nasai. 
It was said that Al-Aqsa Mosque   existed before Sulayman (peace be upon him) and that Sulayman rebuilt it; this is based on the evidence narrated in al-Sahihayn from Abu Dharr (may Allah be pleased with him) who said: “I said, ‘O Messenger of Allah, which mosque was built on earth first?’ He said, ‘Al-Masjid al-Haram [in Makkah].’ I said, ‘Then which?’ He said, ‘Al-Masjid al-Aqsa.’ I said, ‘How much time was there between them?’ He said, ‘Forty years. So wherever you are when the time for prayer comes, pray, for that is the best thing to do.’” (Narrated by al-Bukhari, 3366; Muslim, 520)
The Prophet (peace and blessings of Allah be upon him) was taken on the Night Journey (Isra) to Bayt al-Maqdis (Jerusalem), where he led the Prophets in prayer in this blessed mosque. 
Allah says (interpretation of the meaning): 
“Glorified (and Exalted) be He (Allah) [above all that (evil) they associate with Him]
Who took His slave (Muhammad) for a journey by night from Al-Masjid Al-Haram (at Makkah) to Al-Masjid Al-Aqsa (in Jerusalem), the neighbourhood whereof We have blessed, in order that We might show him (Muhammad) of Our Ayat (proofs, evidences, lessons, signs, etc.). Verily, He is the All-Hearer, the All-Seer.” [al-Isra 17:1]
Who built the Dome of the Rock?
The Dome of the Rock was built by the caliph ‘Abd al-Malik ibn Marwan in 72 AH. 
It says in al-Mawsu’ah al-Filastiniyyah (4/203): 
“The name al-Masjid al-Aqsa was historically applied to the entire sanctuary (al-Haram al-Sharif) and the buildings in it, the most important of which is the Dome of the Rock which was built by ‘Abd al-Malik ibn Marwan in 72 AH/691 CE, which is regarded as one of the greatest Islamic historical buildings. But today the name is applied to the great mosque which is situated in the southern part of the sanctuary plateau.” 
It also says in al-Mawsu’ah (3/23): 
“The Dome of the Rock is situated in the middle of the plateau of al-Masjid al-Aqsa, which is in the southeastern part of the city of al-Quds (Jerusalem). It is a spacious rectangular plateau which measures 480 meters from north to south, and 300 meters from east to west. This plateau occupies approximately one-fifth of the area of the Old City of Jerusalem.” 
Difference between al-Aqsa and the Dome of the Rock 
The mosque which is the place of prayer is not the Dome of the Rock, but because pictures of the Dome are so widespread, many Muslims think when they see it that this is the mosque. This is not in fact the case. The Mosque is situated in the southern portion of the plateau, and the Dome is built on the raised rock that is situated in the middle of the plateau. 
We have already seen above that the name of the mosque was historically applied to the whole plateau. 
This is supported by the words of Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) in Majmu’at al-Rasail al-Kubra, 2/61: 
“Al-Masjid al-Aqsa is the name for the whole of the place of worship built by Sulayman (peace be upon him). Some people started to give the name of al-Aqsa to the prayer-place which was built by ‘Umar ibn al-Khattab in front of it. Praying in this prayer-place which ‘Umar built for the Muslims is better than praying in the rest of the mosque, because when ‘Umar conquered Jerusalem there was a huge garbage dump on the rock, since the Christians wanted to show their scorn for the place towards which the Jews used to pray. So ‘Umar issued orders that the filth be removed and he said to Ka’b: ‘Where do you think we should build a place of prayer for the Muslims?’ He said, ‘Behind the rock.’ He said, ‘O you son of a Jewish woman! Are you influenced by your Jewish ideas! Rather I will build it in front of it.’
Hence when the imams of this ummah entered the mosque, they would go and pray in the prayer-place that was built by ‘Umar. With regard to the Rock, neither ‘Umar nor any of the Sahabah prayed there, and there was no dome over it during the time of the Rightly-Guided Caliphs. It was open to the sky during the caliphate of ‘Umar, ‘Uthman, ‘Ali, Mu’awiyah, Yazid and Marwan… The scholars among the Sahabah and those who followed them in truth did not venerate the rock because it was an abrogated qiblah… rather it was venerated by the Jews and some of the Christians.” 
‘Umar denounced Ka’b al-Ahbar and called him the son of a Jewish woman because Ka’b had been a Jewish scholar and rabbi, so when he suggested to ‘Umar that he should build the mosque behind the rock, that was out of respect for the rock so that the Muslims would face it when praying, and veneration of the rock was part of the religion of the Jews, not the religion of the Muslims. 
The Muslims’ fondness for the picture of the Dome may be because of the beauty of this building, but this does not excuse them from the resulting mistake of not distinguishing between the Mosque and the buildings that surround it. 
This may be one of the plots and tricks of the Jews, because of their veneration for the rock and their facing it in prayer. Or it may be in order to give importance to the Rock so that they can fulfil their desire to build the so-called Temple of Solomon on the ruins of al-Masjid al-Aqsa. This is by making the Muslims think that al-Masjid al-Aqsa is the Dome of the Rock, so that if the Jews start to destroy al-Masjid al-Aqsa and the Muslims denounce them for that, they will tell them, “Al-Masjid al-Aqsa is fine,” and will show them a picture of the Dome of the Rock. Thus they will achieve their aims and be safe from the Muslims’ criticism. 
We ask Allah to restore the Muslims’ power and glory, and to cleanse al-Masjid al-Aqsa of the criminals, "for Allah has full power and control over His Affairs, but most of men know not." (Surah Yusuf 12:21)
And Allah knows best.