A lot of it has to do with how much impact the Muslim Brotherhood has had on each respective country. The Muslim Brotherhood is mostly seen as a Muslim political movement
The Muslim Brotherhood is a missionary movement founded in Egypt in
1928 by Hassan al-Banna, a schoolteacher working in the town of
Ismailia, near the Suez Canal. He argued that an Islamic religious
revival would enable the Muslim world to catch up to the West and
shake off colonial rule.
But he was sweeping and contradictory about the mission of the group,
and largely avoided spelling out what an Islamic government might look
like.
It's that vagueness about goals that creates the controversy. In particular, the Brotherhood generally advocates for Sharia law
Sharia is a body of religious law that forms a part of the Islamic tradition. It is derived from the religious precepts of Islam and is based on scriptures of Islam, particularly the Quran and the Hadith.
Egypt
A lot of the reason why Egypt classifies the Brotherhood as a terrorist group has to do with their former president, Mohamed Morsi, a man who was elected almost entirely thanks to the Brotherhood.
In November 2012, Morsi issued a proviso constitutional declaration that granted him unrestricted authority and the authority to legislate without the need for judicial oversight or review. This was a move to stop the Mubarak-era judges from getting rid of the Second Constituent Assembly. The new constitution that was then hastily finalized by the Islamist-dominated constitutional assembly, presented to the president, and scheduled for a referendum before the Supreme Constitutional Court could rule on the constitutionality of the assembly, was described by independent press agencies not aligned with the regime as an "Islamist coup". These issues, along with complaints of prosecutions of journalists and attacks on nonviolent demonstrators, led to the 2012 protests. As part of a compromise, Morsi rescinded the decrees. A new constitution was approved by approximately two-thirds of voters in the referendum, although turnout was less than a third of the electorate.
While some of Morsi's actions were un-democratic, a lot of the underlying controversy came from the fact that he (and the majority-Brotherhood Egyptian Assembly) were seen by many Egyptians as attempting to enact Sharia law by rewriting the Egyptian Constitution
Much of the international concern about the Islamic sharia in Egypt stems from the growing political role of the Muslim Brotherhood and the party it founded, the Freedom and Justice Party. The Brotherhood has confused many (including, at times, its own members) by a variety of statements and proposals on the Islamic sharia. The movement’s confusing position stems from two impulses that pull its members in different directions.
It's not clear the Brotherhood directly fomented any violence, but Morsi's actions seem to have given the Egyptian government room to condemn his political affiliations as well.
Israel
The Muslim Brotherhood doesn't really have a lot of room to do much in Israel. Only about 20% of the population living within Israel (i.e. the 1967 borders) is Arab/Muslim. The other problem is that the remainder of Muslim politics within Israel's borders is already dominated by groups like Fatah and Hamas (with Hezbollah to the north in Lebanon).
One rare area where the Botherhood gets any mention would be the Islamic Movement in Israel, which has split over political engagement in Israeli politics
The Islamic Movement in Israel split into two branches in 1996 over the question of whether to stand candidates for election in the Israeli Knesset. The southern branch chose to engage in the Israeli political system and formed the Ra’am party. By contrast, the northern branch chose to boycott the political system and elections. Today the Northern Branch is led by Raed Salah. It is thought to have relations with the Muslim Brotherhood and Hamas.
In November 2015, the Israeli government issued an executive order under the 1945 Emergency Regulations that outlawed the northern Islamic Movement and 17 associated NGOs. The Murabitat and Murabitoun, a collective established by the movement to ‘defend’ Al-Aqsa, were also outlawed. The deputy leader of the Northern Branch, Kamal al-Khatib, was arrested by Israeli security forces in May 2021 and subsequently indicted for “incitement to terrorism, violence, and identification with a terrorist organisation” in relation to widespread protests by Palestinian citizens of Israel. Raed Salah has also been arrested on several occasions.
My bet is the terrorist part of the movement gets lumped in with all the other Hamas activity (which is pretty prolific already). This means the Brotherhood is merely backing a minority party in the Knesset. Trying to ban it might mean the removal of the Arab minority.
