Well First lets talk about what the Berlin Conference was. It was an attempt to avoid war between European powers by codifying how African Colonisation worked. There'd been something of a rush on the continent, in early 1884, you had German Warships literally just ordered to head to Africa, find any bit that nobody had else claimed and get someone off the boat. The hope was by formalising it and laying out areas of interest which the other powers would respect this would avoid potential war.
The Conference would not have happened without the USA. The American's were also worried about a war starting in Africa and they wanted central Africa to be a neutral zone with no fighting and a free trade zone with no tariffs. German and African traders often dominated trade with non conquered African countries but once an Empire took it, they would raise tariffs and drive them out. That's what the British did at Lagos.
In April 1884 President Chester Arthur of the USA recognised Leopold's International African Organisation as the rightful government of the Congo basin. This would ensure a neutral free trade zone. The Germans reacted by throwing their weight behind this plan and holding a conference to sort out various claims. During WWI the Germans claimed that the USA had a responsibility to protect German East Africa from British attacks as they had viewed the americans as the guaranteer of the agreement of the berlin conference for no fighting between europeans in the free trade zone. So the USA was seen as an important figure, they had a unique status over the Berlin agreement in terms of being able to reject it, they were seen as the power that guaranteed African peace and could enforce the contract. Likewise giving the valuable Congo basin to a neutral power and various terms of free trade among the niger basin and central africa were attempts to remove friction between different colonial emprise and avoid war.
The Berlin Conference didn't really work. No European wars started in Africa (the closest it came was either the Fashoda incident between UK and France or the Agadir crisis between France and Germany) but ww1 was still fought in Africa and post of the free trade agreements weren't honoured in practice. Areas of influence were established but it was also agreed that claims were worthless unless you had boots on the ground. So as a result there was a rush to turn areas of influence into actual control even in areas given to somebody else. It didn't really achieve anything in terms of changing pre Berlin patterns and preventing conflict.
Liberia and Ethiopia are arguably the only two exceptions to this where Berlin actually mattered. In that they were saved by it.
I'll look at Liberia first. The American-African settlers arrived in Monrovia in 1822. This idea of a country for African Americans to be sent to and thus end racial tension in America was very popular among a lot of white Americans and had at various points the support of the American Government, who viewed Liberia as some kind of responsibility of theirs and thus to an extent conquering Liberia would be risking the treaty by angering it's guaranteer. In late 19th Century, the diplomatic viewpoint of Liberia was it was only a matter time before it became a formal colony of the USA to protect it from the Europeans.
After all large areas of what was than Liberia was colonised by Europeans. Because of the agreement that claims were worthless unless you had boots on the ground, the Europeans argued that Liberia did not control all of its claimed land and just annexed parts of it. Liberia in 1869 claimed to rule 180,000 square miles of territory but by 1900 that was only 43,000 square miles. The rest was seized by France, in three different treaties, and the UK, in one.
The Rump Liberia survived but only just. Germany in 1898 send a gun ship to Liberia to take control of it and only backed down when the British took the Liberian side. Likewise the French had plans to completely take Monrovia, and the British sponsored an attempted military coup there to overthrow the government which failed. But the attempts were largely secret and backhanded because the USA was willing to send messages to European embassies saying they viewed Monrovia as their allies and from 1912 onwards posted US soldiers in the country. That relationship combined with rivalry between the colonial powers is probably what prevented an actual invasion.
Ethiopia was less lucky. It was invaded in 1896 by the Italian Empire.
At the Berlin Conference that area of the World was seen as an Italian area of interest. And they acted on it by signing the 1889 Treaty of Wuchale which, in the Italian version, signified that Ethiopia would become a vassal state of Italy (the Amharic version did not say this and Ethiopia angrily denounced this once they realised that). So France and the UK weren't going to invade Ethiopia because they viewed it as a part of the Italian Empire. And this is the build up to WWI, getting the Italians on your side in a potential European War was viewed as crucial. Ethiopia wasn't worth alienating the Italians.
Italy, on the other hand, did invade to establish their control over the area they had claimed and well they lost. Ethiopia had a bigger and equally well armed army with better generals and better trained men. This was not unique, African Armies won plenty of other battles against Europeans during this time period but the other times the Europeans came back with more men within 15 years and then won. In Ethiopia it was 40 years later that the Italians came back and did actually conquer it. Ultimately the political will in Italy wasn't there, it was one of the poorest of the major powers involved at the Berlin conference and the loss was seen as the fault of the government and it was deemed that the country couldn't afford a full campaign to conquer a well armed foe like Ethiopia. The cost of the successful 1930s conquest of Ethiopia was arguably one the Italian state couldn't afford.
So the answer in short is Liberia survived because it was allied to the USA and the USA was the guaranteer of the document agreed at the Berlin conference. And Ethiopia survived because at Berlin it was placed within the Italian sphere of influence and the Italian Army was inferior to the Ethiopian Army and so could not conquer it.