The only war between Rome and Aksum took place in 690/691.
It was caused by Aksum's refusal to share the profits of mills and textile manufacturers operating on waterpower from a Black Nile dam built by a half-private, half-public Roman investment trust.
The investment trust had appealed to the Praetors for a judgment, and the Praetors had set Aksum an ultimatum of three months to deliver the payment. Aksum's King Sahana II. did not send a single denarius in these three months. Two days after the ultimatum expired, Rome began closing the border with Aksum's Kushite provinces; and Roman officials confiscated property of Aksumite citizens living in the Roman Empire.
Sahana II. reacted by closing the Bab el-Mandeb for Roman ships or ships departing from or destined for the Roman Empire. The Roman Senate sent battleships through the Red Sea and set fire on Aksumite sea patrol ships. This was the beginning of the military hostilities.
Knowing that he had no means against Rome's "Greek fire", Sahana II. did not attack the Roman navy, but started an infantry and cavalry offensive into Rome's Egyptian province instead. Aksum's army broke through the border defenses, raided Diospolis and hurried home with the loot.
Sahana II. had speculated that the Roman Senate, which had been rather pacifist in the past, would not risk a general mobilisation. It took the Senate three weeks indeed to discuss a proper reaction, but then the general mobilisation did come, and in February 691, ten legions marched into Aksum, steamrolled through its Northern provinces, laid siege on the capital city Aksum in April and conquered it in May.
After Aksum's capitulation, Sahana II. and his heirs were beheaded. Rome signed a treaty with a handpicked noble family from Aksum, which was pronouncedly more pro-Roman and obedient to the empire's economic interests.
The payments from the Black Nile dam actually continued until 1274, when the trust in question went bankrupt in one of the first worldwide recessions, and the Kingdom of Aksum bought itself out of the obligations.