All praise is due to Allah, and peace and blessings be upon the Messenger of Allah.
Both of these statements are authentically reported from the Prophet ﷺ, and there is no contradiction between them and the other texts of the Qur'an and the Sunnah. The methodology of Ahl al-Sunnah is to reconcile all the textual evidence rather than accepting some texts while ignoring others.
"Whoever has an atom's weight of faith in his heart will not enter Hell."
The meaning of this hadith is that a person who possesses faith will not remain in Hell forever as the disbelievers do, even if he enters it because of major sins for which he died without repentance.
A sinful believer may be forgiven by Allah from the outset, or Allah may punish him according to his sins before bringing him out of Hell through His mercy or through the intercession of those permitted to intercede. Therefore, no one who has even the smallest amount of faith in his heart will remain in Hell eternally.
This is explained by the hadiths of intercession, which state that sinful monotheists will eventually be taken out of Hell, while eternal punishment is reserved only for the disbelievers.
"No one who has an atom's weight of arrogance in his heart will enter Paradise."
This does not mean that everyone who falls into arrogance will remain in Hell forever or be permanently deprived of Paradise. Rather, it means that such a person will not enter Paradise immediately with those who enter it first if he deserves punishment. He may first be punished and then admitted into Paradise, or Allah may forgive him from the outset.
The Prophet ﷺ himself explained arrogance by saying:
"Arrogance is rejecting the truth and looking down on people."
That is, refusing the truth out of pride and despising or belittling others.
It is not arrogance for a person to like wearing beautiful clothes, shoes, or maintaining a good appearance, for Allah is Beautiful and loves beauty.
Ahl al-Sunnah reconcile the texts of promise and warning by affirming that:
In this way, all the Islamic texts remain consistent without contradiction.
The hadith "Whoever has an atom's weight of faith in his heart will not enter Hell" means that such a person will never remain in Hell eternally like the disbelievers, even if he enters it because of his sins.
The hadith "No one who has an atom's weight of arrogance in his heart will enter Paradise" means that if he deserves punishment, he will not enter Paradise immediately until he has been purified from his sins or Allah forgives him.
This is the belief of Ahl al-Sunnah wa al-Jama'ah regarding the reconciliation of the texts of promise and warning.
And Allah knows best.