From the sources, it seems that Harun (a.s.) was older than Musa (a.s.).
And Allah knows best.
When Prophet Haroon (AS) was alive, he was the deputy of Prophet Musa (AS), and was the partner of Musa in his responsibility and real helper and supporter of Musa, but he passed away during the life of Musa (AS). Had he lived after the passing away of Musa, no one was entitled to succeed Musa but Haroon.
The status of Haroon from Musa is similar to the status of Ali (AS) from Muhammad (SAWA). (Saheeh al-Bukhari; Hadeeth numbers 3430 and 4064.) and Saheeh Muslim; Hadeeth numbers 4416, and 4419 and 4420 and 4421.
Thise understand Quran can obviously see the status Ali (AS) from the Prophet (SAWA) which is superior to all Muslim Ummah.
There are many similarities between Prophet Haroon (AS) and Ali ibn Abi Talib (AS).
1. Prophet Haroon (AS) was the brother of Prophet Musa (AS) while Ali (AS) was the cousin brother and spiritual brother of Prophet Muhammad ( (SAWA) who said to Ali : O Ali, You are my brother in this life (Dunham) and hereafter (Aakhirah). (Al-Mustadrak 'Ala al-Saheehain by al-Haakim al-Nisabori, Hadeeth number 4345).
2. Prophet Haroon (AS) was the partner of Prophet Musa (AS) وأشركه في أمري)( Make him my partner in my responsibilities) in conveying the message of Allah to people and facing the hardships from the enemies, while Ali (AS) was the partner of the Prophet Muhammad (SAWA) in his responsibilities in conveying the message and protecting it.
3. Prophet Haroon (AS) was the main helper of Prophet Musa ( AS) وزيرا, while Ali (AS) was the main helper of the Prophet Muhammad (SAWA).
4. Prophet Haroon was the leader of the followers of Musa when used to leave them, اخلفني في قومي ( Be my deputy in my followers) while Ali (AS) was the deputy of the Prophet Muhammad (SAWA) who declared : من كنت مولاه فعلي مولاه For whom so ever I am the leader, Ali is his leader (al-Tareekh al-Kabeer by al-Bukhari, V. 1, P. 375 and V. 4, P. 193) also ( Musnad Ahmad Ibn Hanbal, Musnad al-Asharah al-Mubashshareen, Hadeeth numbers 606 and 633 and 906 and 915 and 918 and 1242.
5. Prophet Haroon (AS) suffered from deviation of many of the community of Prophet Musa (AS) when he left them, while Imam Ali (AS) suffered the same and more from those Muslims who deviated from the teachings of the Prophet (SAWA) immediately after him.
6. Prophet Haroon had three sons; Shubbar, Shabbir and Mushabbar while Ali ( AS) had from Lady Fatimah (AS) three sons; Hasan, Husain and Mohsin. These names is similar in Arabic to the names of the sons of Prophet Haroon.
7. Prophet Haroon was tortured by some of the community of Prophet Musa (AS) وكادوا يقتلونني while Imam Ali (AS) was tortured and was near to be killed by the Saqeefa gang to force him to give allegiance to them. He was ultimately killed by Ibn Muljam.