There are two reasons why nations with differing beliefs and opinions are forced to coexist.
1 They each have something the other one needs
If both of the nations produce something that the other one needs, then they are both forced into a standstill. They can't destroy the enemy nation because that would mean giving up on the products that they produce.
For the sake of simplicity we will call
Nation A-the Authoritarian Regime
Nation B-the Democratic State
Nation A has horrible civil rights abuses, but so many people work in its factories that it has vastly superior manufacturing compared to Nation B. In particular, it specializes in making weapons, specifically guns and ammunition, among other things.
Nation B is a great place to live in, but it can't compare to the sheer industrial might of Nation A. It is horrible at making its own weapons. However, it is much better at agriculture. It is great at making and producing bread among other food items.
Now, because Nation A is great at making weapons and Nation B is great at making bread, they make a trade-off. Nation B gives A food to feed their starving people, and Nation A gives B the weapons to properly defend their homeland.
This is why they can't ever go to war against one another. They rely on each other too much.
If A decides to get a little too trigger-happy and considers attacking, B will just withhold its bread supply from them and their people will slowly starve.
If B ever decides it wants to go to war, A will just stop trading with them and cripple A's economy in a similar fashion to the above example.
You can't start a war if you don't have the guns to wage it.
This is one reason why two nations with diametrically opposed views would decide not to attack each other. Mutual need.
No matter how much Nation B might hate the civil rights abuses in A, what are they supposed to do if fighting them would cripple their economy? Would it even be worth it?
2 They are scared of the power the other one possesses.
If a mutual need is not the driving factor of this relationship, then mutual fear is more than able to do the trick.
Even if both nations absolutely hate each other's guts, they know what would happen if they went to war with one another. Both countries have armies and weapons stores that are formidable enough to make each other pause before they consider an all-out war.
Think of the United States and the Soviet Union during the Cold War. Those two hated each other, but neither one was really able to do anything about the other because of the threat of mutually assured destruction at the hands of nuclear weapons.
Fear is a powerful motivator.
Obviously, the people of Nation B would like it if the people of Nation A were no longer under the rule of an oppressive dictator, but if that dictator has any number of nasty weapons, they would have to be crazy to even set foot on their soil.
Whether they like each other or not, ambassadors of both countries will have to grit their teeth and smile for the cameras, because they know that only one small fuse is all it would take to burn both their countries to the ground in the flames of war.
This assumes Nation B even cares about the tyrants in the first place.
Just because a country is a democracy does not mean it's a perfect one. It could be flawed to the point where it is just as bad if not worse than an outright autocracy. It can be rife with greed, lies, and countless other abuses of power. Honestly, this is common in all governments, no matter how outwardly nice they may seem.
Why would a corrupt government official care about the tyrant of a nearby nation-state? They might see each other as rivals, but they might not even care about the civil rights abuses going on in the rival nation, especially if they were committing similar atrocities within their borders.
Countries make such moral compromises all the time.
Even if the country doesn't want to turn a blind eye, what are they supposed to do? Are they just supposed to ignore the sovereignty of another nation and charge in guns blazing, telling them to completely change their government structure? This isn't a black and white moral quandary. Forcing a foreign nation to accept democracy, for example, is not democracy. It's more like the prelude to a bloody war.
Mutual need and economic trade forces two nations that might not like each other to work in tandem, or else suffer.
Mutual fear drives two nations that hate each other to hesitate before they go to war and end up annihilating each other.
Lastly, mutual corruption makes it such that immoral political figures will easily turn a blind eye to civil rights abuses.
Even if the relationship between these two nations is rocky, they just have to deal with it in one way or another.