Just my two cents:
The current state of affairs regarding Radical Islam is the end result of a series of events promulgated and instigated by America itself. Starting with the U.S' toppling of Mohammad Mosaddegh's democratically elected government in Iran, which set the stage for the Iranian Revolution, which set the stage for the Iran-Iraq war, which set the stage for the Gulf War to follow. These events and the powers behind Iraq and Iran destabilized the region and resulted in the chaos we see today.
Sunni groups rally against America for the invasion of Iraq: An invasion that would not have occurred had America either allowed Mosaddegh to continue his rule, or had Bush Jr. listened to anybody but his Neoconservative warhawks prior to initiating an invasion on false pretenses (this matter was worsened by the utter lack of anything resembling a plan in rebuilding Iraq after the invasion). Shi'ite groups rally against America for the deposition of Mosaddegh, and American support for Iraq during the Iran-Iraq war. In essence, both Sunni and Shia populations are entirely within their "right" to hate America, because this is technically America's fault, so in asking these specific nationalities to "let it go" regarding their hatred for America...you are in essence asking a very difficult thing of an extremely disenfranchised populace.
I firmly believe that if the West wanted anything but this state of affairs, then it was the responsibility of the United States and the nations which contributed to this chaos (the UK via deposing Mosaddegh, Sykes-Picot, failed colonial legacies in Bahrain, Iraq, Palestine; France because goddamnit France has been terrible to Muslims generally for the past century and I shouldn't have to list this one) to have either left behind states and institutions in the Middle East that were not completely flawed and mis-structured (they were), or at the very least taken in refugees from said nations when they inevitably fall into chaos to alleviate their resource issues and ethnic tensions in the wake of civil wars and invasions that they themselves instigated.
There is therefore a burden of responsibility on certain Western nations to take in Syrian/Iraqi/Yemeni refugees, not just from causing these events in the first place, but from refusing to deal with the aftermath in any rational and reasoned way and from instigating the chaos further with every mistake and terrible policy decision made by Bush and Obama.
Unfortunately, there is a lot more chaos in the world. Some which the West did have a large hand in instigating, some where it did not instigate but instead propagate, and some where the populace itself has just continuously shot itself in the foot. Refugees from Africa, Central Asia, and other Middle Eastern countries are flooding in through the open gateway provided by the Syrian Refugee Crisis. The end result of this action is not terrorism; there is very little to no terrorism actually committed by refugees themselves. The end result is actually a higher crime rate, a budgetary crisis in what was already a stressed social services system (the 2008 financial crisis really gently-carressed up Euro budgets), and a possible rise in rape rate among European countries who've been flooded by said refugees. It's extremely doubtful that all of these refugees fleeing by sea or land to Europe are really war refugees to begin with, and their departure contributes to a decline of society both in their home countries and in the countries they end up in.
Essentially, while it is some of the West's responsibility (America's, France's, the UK's) to take in refugees from the Middle Eastern countries they gently-carressed up and refused to fix, it was not Germany's or the Scandinavian countries' responsibility to take in millions of people from Africa or Central Asia that are only actually fleeing low incomes and poverty in their messed up countries. Thus said Euro countries have paid the obvious price of allowing a massive unskilled and uneducated labor force into their countries. Their open door policy was well-intentioned, but the crisis could have been handled much better than the unorganized chaos and wanton abuse of refugee status that persists until today. The refugee status exists for a reason, and labor drain from African nations is not one of them.
This is why people have such a mixed view of refugees (aside from the obvious terrorism fearmongering). The countries actually taking them in are not the ones who caused the problem in the first place, the people coming in are not the ones who are supposed to be coming in, and the people who are coming in to these countries are a massive unskilled labor force with little connection to German or Scandinavian tradition. This kind of massive immigration is unheard of in any time prior, and it threatens to diminish the core culture that exists and has existed within said European nations for thousands of years.
Trump's policy does not take into account any of these factors: He does not care about America's responsibility, and this is reflected in the policy's wording.
Trump's policy is not affected by any of these conditions: Muslim Refugees coming in to America have never committed a single act of terrorism on U.S soil, and America (by virtue of the Atlantic Ocean) has the ability to slowly process and verify that refugees are actually reasonable and useful Syrian/Iraqi/Yemeni refugees (this isn't as unlikely as you think: Syria, Iraq, and Yemen had a decent economic/social system once upon a time, and they're generally good people). American culture is multifaceted by virtue of being an ethnic diaspora and already has a sizable Syrian, Somali, Yemeni, and Iraqi community within it that happens to be going along very nicely with the local populace (barring the occasional defamation/arson/graffiti/shooting of a mosque): A few thousand refugees aren't going to dilute American culture in any measurable way.
Trump's policy does nothing but limit the U.S' culpability for its own actions, throw a massive gently-carressing wrench into the lives of hundreds of thousands across the world who were otherwise perfectly useful U.S residents or future residents (Iranian professors, Somali students, Iraqi translators who worked with the U.S army, etc...).
This ban is a conception of Trump's. It has no relation to Obama's prior policy in anything but a cosmetic sense, and the news is lying to you by drawing a similarity between them. The ban and the accompanying hard limit on refugees are a travesty of idiocy made to show the people that Trump is somebody that takes action quickly: It has no reason to exist within American society, and it does not make any logical sense beyond rampant paranoia given that existing vetting processes are clearly working in preventing extremists from reaching American soil.
TL;DR: I will not post a TL;DR. TL;DRs are the reason why nobody understands this problem at all: Every single facet of the problem that I posted is equally important and cannot be summarized without trivializing the issues at work.