I see several things happening to Abyssinia:
1. For one this almost certainly means the end of the position of Emperor. The prestige of the Neguse Negest was largely due to the policies of centralization and conquest carried out by Tewodros II and Menelik II. Assuming a POD around the Battle of Adwa, Menelik will still be Emperor and all of that work will be undone. Even when region gains independence, the line of Solomon won't be there to lead it when it does.
2. The Italians will try to play the major ethnic groups off of one another, so this may lead to a lot more infighting between Oromo, Amhara, and Tigrinya than even takes place in OTL today. It may also mean smaller ethnic groups like the Afar and the Beta Israel will get thrown under the bus.
3. Ogaden will not be administered as part of Abyssinia nor will it be a part of Abyssinia come independence. Being largely Somali the Italians will administer it as part of their Somalian territories. The Ogaden even being considered part of Abyssinia is largely due to the conquests of Tewodros II and Menelik II, so there isn't much of a precedent for that region to see itelf as Ethiopian.
4. There will be attempts by the Roman Catholic Church to make inroads in Abyssinia on a scale that hasn't been seen since the Emperor Susenyos. Its success will be mixed, but Catholicism will quickly be associated with the colonizers and Ethiopian Orthodoxy and Islam with the colonized. This could interestingly enough lead to a more secular Ethiopia, as opposition to Catholicism might drive the Orthodox and Muslims together, at least long enough to gain independence. Something like that may even be strong enough to mitigate the strife caused by (2).
5. There are a number of things about Ethiopian culture that will be suppressed by the Italians for the sake of administration. For example, the Ethiopian calendar is about 7 years behind the West's and has 13 months, and when an Ethiopian tells you it's 6 o'clock they mean it's midnight (and vice versa). I can see the latter completely disappearing from Abyssinia, though the former will almost certainly continue to exist in a religious context.
6. Amharic, Oromo, Tigrinya and the various Ethiopian languages will have a lot more Italian loanwords, though I don't see Italian cuisine having too much of an influence. Ethiopian food is too tied in to both the region and culture of the region for that to change substantially. However I could see the Ethiopian coffee ceremony become more popular in Italy.