This is a complex question. Here's an answer to it, as far back as I could go in order to explain the shared heritage between the states and the rivalry that followed. For reference - Habesha is a term used to describe some, not all, of the peoples who inhabit Ethiopia and Eritrea. They speak Afroasiatic languages, write (mostly) in the Ge'ez script, and are unique to the Horn region. The main groups of note in this case are the Amhara and Tigray of Ethiopia, and the Tigre and Tigrinya of Eritrea.

Aksum to Medri Bahri

The Kingdom of Aksum was the first verifiable Habesha state and occupied North Ethiopia and Eritrea, as well as Yemen. This was during the Classical period, with Kingdom a contemporary to Rome. It was during this time that the region, in particular the Highlands, became Christianised. While there aren't many primary sources, we know that Aksumite Kings like Kaleb and Ezana were in contact with the Byzantine Empire, and warred in pre-Islamic Arabia.

Things get especially murky following the rise of Islam. We know that Aksum lost it's holdings in Yemen to the Sassanids and the Sabaeans, but not much else is known apart from that. It fell around 940CE. After this, the Zagwe ruled in Ethiopia, and were succeeded by the Solomonids, while in Eritrea the state of Medri Bahri formed.

Medri Bahri was at times a vassal to Ethiopia but was almost always a distinct political entity. It was ruled by the Bahri Negus, the "Sea-King," and this is where what can be perceived as tensions start. Negusa Nagast (Emperor, lit. King of Kings) Zera Yakob stated in his chronicle that he appointed the Bahri Negus to his office, through his right as the Suzerain of the area. Of course, this was done through coercion in order to make the Imperial ally the strongest of the local polities in the area. Zera Yakob then instituted a military colony in the state, effectively trying to start a military occupation of Medri Bahri. Following this, some 40 years later in the 1520s, Portuguese explorers note that the current Bahri Negus was the uncle of Emperor Lebne Dengel and paid him tribute. However, Medri Bahri also joined with the Ottomans against the Ethiopian Empire in 1572

You can see the attitude Medieval Ethiopia took to Medri Bahri from this. Ethiopia viewed Medri Bahri as a vassal that was more or less part of their Empire, whereas Medri Bahri was a distinct entity. This continues for several centuries - James Bruce visited Ethiopia in 1770, noting that the state (undergoing a period of civil strife known as the Zemene Mesafint) was frequently at war with it's northern neighbour.

The Imperial Period

Modern Ethiopia began with Menelik II. His predecessor, Tewodros II, had united country but had died facing off with the British in Magdala during the Abyssinian Expedition of 1868. After Tewodros came a brief resumption of conflict after a man named Tekle Giyorgis proclaimed himself the Negusa Nagast of the restored Zagwe Dynasty, but was overthrown and replaced with Yohannes IV. Yohannes was a complex man but undoubtedly started Ethiopian foreign outreach, cooperating with the British in Sudan against the Mahdi. He is also pictured here in this montage of world Heads of State, the first from the left.

When Yohannes died, Menelik succeeded him. While the Empire that Yohannes ruled was a small, but unified, state, Menelik expanded to the borders of Modern Ethiopia. One of the states he sought in his expansion was Medri Bahri. He succeeded in incorporating it by force, but it was taken from him by the Italians after the Congress of Berlin. Ethiopia remained independent, but Eritrea (as the Italians named Medri Bahri) was part of the colonial machine. But the Italians wanted more.

The first Italian-Ethiopian War ended with the spectacular defeat of the Italians at Adwa and cemented Menelik's already incredible reputation as a leader. Ethiopia did not press on to Eritrea, however. This resulted in Italian cultural influence (though, it must be said, extremely minor influence) beginning in Eritrea. The Italians built a railway and some buildings, but that was about it. BUT Eritrea was it's own distinct entity once again, even more distinct now that it wasn't even independent.

Haile Selassie

Emperor Haile Selassie is perhaps the most famous of Ethiopia's Emperors. After WW2, he was triumphant, his country liberated, and, eventually, he was the face of Africa's decolonisation movement (next to Gamal Abdel Nasser of Egypt and Kwame Nkrumah of Ghana). Eritrea, meanwhile, was now a possession of Britain. In time, decolonisation came to Eritrea. This was seen as a chance for the people of Eritrea to finally have their own independence - a good thing for all involved.

Britain then gave Eritrea to Ethiopia.

This, arguably, is the immediate cause. The state had been, in the eyes of the now-large Eritrean Independence Movement, denied it's freedom too many times. Haile Selassie, the great liberator and face of Africa's future, was an Autocrat at heart. Saudi Arabia is, undoubtedly, an absolute monarchy - but not of the sort that Ethiopia was at this time. Ethiopia was more like a feudal kingdom, completely out of time, and the Emperor appointed Viceroy's to Eritrea rather than represent the people there.

As well as this, local officials would often try to misrepresent how well their region was doing. When the Emperor would visit provinces, Governors would only take him to the best parts, show him how well some parts were doing. The Army was used to crush revolts.

Centuries old tradition meant that the people had to prostrate themselves before the Emperor. Power was measured by how close you were to the Emperor's ear. One powerful figure was the Minister of the Pen, the man in charge of writing down the decrees of the quiet, soft-spoken Emperor.

Then, in 1960, there was a coup attempt by some of the Emperor's circle while he was in Brazil. The Royal Guard proclaimed his son, Amha Selassie, the new Emperor. The coup was crushed, but the writing appeared on the wall. With this in mind, the Eritrean Liberation Front began it's 30 year campaign.

Within 14 years, the rest of the Empire would devolve into civil war between the Marxist Derg and the coalition of opposition groups that would become the EPRDF.

Independence and War

After the end of the Civil War, Eritrea was granted independence under Isaias Afeworki. Eritrea and Ethiopia disputed the demarcation of the border from 1991, until May 1998 - when Eritrean officers were killed in the disputed Badme region. Eritrea sent a mechanised force into the region, beginning a brutal war that lasted 2 years.

Since then, the tensions have long been apparent. Luckily, PM Abiy Ahmed has seen a thaw in relations with Eritrea - the border is open again, and a peace treaty has been signed. Things are starting to look up for both the countries relations, though both are still plagued with problems. In Ethiopia, previous Governments since 1991 have largely used force to resolve Ethnic problems, which very well could spiral into another conflict. Eritrea is a one party state, and even with this rapprochement from Ethiopia is still very isolationist.

I hope this answers your question.