You've revised the question in comments; you're not asking why people opposed Communism, you're asking why they opposed it more than Fascism (dictators)
The difference is twofold - first in the claims they made and second in the cultures where they took root.
What claims did Communism make?
Communism advocated worldwide revolution, the abolition of private property and the establishment of a new world order. (pick your own source - they all make me sick to the stomach to read)
According to the communists, only the laboring class would rise; all others would fall (and probably fall to the knife). Take a quick look around you - are you willing to bet your family's life that you're one of the elect? Are you completely sure that there isn't someone poorer than you who will claim that you are bourgeois? (Hint: if you're using the internet, I can guarantee that there is a class of people who will happily denounce you, and seize your property as their own.)
Communists offered class solidarity to a world that wasn't accustomed to thinking in terms of classes. (I yearn for those halcyon days when we could actually think about a problem before first assigning blame to the bourgeois). You would ultimately be united with people who shared your class interest, but you had to learn communism before you understood your class interest. And of course if you didn't understand your class interest, you were a counter-revolutionary and should be punished. Is there anyone who wants to admit that they don't share their class interest? Trains to the labor camps form on the left.
Communists rejected religion and pretty much all traditions. If there was something you liked about your childhood - a holiday, a meal, the notion of a birthday present? that was counterrevolutionary and must be suppressed by force. Everything you remember as pleasant must be replaced by communism. You have no immortal soul, there is no God. If you believe there is a god, the train to the labor camp starts on the left. If your parents are religious, the state will ship them off to a labor camp; all you need to do is betray them to the state. Or if you have some loyalty higher than the state, the train to the labor camp begins on the left. _Note - @jamessqf says that anti-religion was a major selling feature of communism. I get that - the Great War had made it tough to believe in a loving God. But there is a big difference between doubting god and never celebrating Christmas again (or Eid, or Yom Kippur, or..). Most of us have a bit more trouble telling our parents that we reject everything they have taught us about good and evil. That not only are they profoundly wrong, but that unless they embrace the revolution, the train to the labor camp begins on the left.
My point is that communism demanded total loyalty. Communism was more important than anything else in your life. Nobody believed that the communist utopia would arrive in your lifetime; the only reward you could hope for was to struggle and die so that the proletariat would eventually rule the world. Remember Boxer?? The only thing holding back the proletarian revolution is that you're not trying hard enough. You - you personally are failing the revolution. You should die of shame, but the party, out of kindness, will work you to death to atone for your crimes.
Also in response to @Jamesqf, that argument carries a great deal of weight with the young - they tend to reject their parents beliefs. But OP is asking about national leaders who aren't that young anymore.
What claims did Fascists make?
Mussolini and Hitler offered to bring order, to bring pride to the nations, to cultivate and raise up the people to their true nature. They promised to keep order, to keep people safe, to enforce the law and make the trains run on time.
There is also a cultural context. Germany and Italy needed hope, and their culture was receptive to ideologies that provided hope. I'm not as familiar with Italian Fascism, but remember that German Fascism was in part a reaction to French Liberalism. France conquered Europe and imposed "liberalism" on the conquered people. Germany knew what "liberalism" looked like, and it looked a lot like oppression. Germany explored alternatives to Napoleonic Liberalism - alternatives in which it was possible to be proud to be a German.
BBC History had a very good article on Italian reactions to Mussolini a couple of years ago - I find it nearly impossible to cite articles from BBC history, but if you search their website you should be able to find it.
Dictators offered racial/ethnic solidarity - you would be united with people like you. People like your neighbors and friends. The power of the nation united would transform you.
Tradition was good - the traditions, habits, cultures and religion of your parents are part of what makes our nation great - you should be proud of these traditions. The future is so bright we've got to wear shades!
I'm not defending fascism; both were evil, anti-liberal ideologies. Both ideologies are effectively religious and require you to believe in things you cannot see. But ultimately Fascism promises order, safety, comfort, unity, and the removal of all danger. Communism promises revolution, the opportunity to fight, to kill, all in service of a future that nobody can describe, except that it is different from today. At the root, both ideologies promise absolution for killing people you fear; because the people you're killing are not really people. They're capitalists, or bourgeois - and therefore killing them is not a crime, it is a noble act in defense of the future. And pretty much anyone you don't like is probably one of those "others".
Edit based on @user2520938 - remember that fasicst wanted a better future for their country/their race/whatever. Communists could not be content with less than the world. Every human being must participate in the proletariat revolution or die.
Fascism was surgical - remove the liberal elements and revive the people. Communism is an ideology of mass destruction.