The main difference between the two members in your last code snippet is encapsulation
See the "Encapsulation" part on this page:
https://learn.microsoft.com/en-us/dotnet/csharp/fundamentals/tutorials/oop
Hiding the internal state and functionality of an object and only allowing access through a public set of functions.
You asked about what private field gets generated if you use an automatic property. In fact the compiler will generate a private backing field usually called something like k__BackingField or similar.
For example, I created a basic class as follows:
public class Dog
{
public string Name { get; set; }
}
This definitely creates a field in the background, and I can find out by using ILSpy to decompile it and we see the following:

Notice the Name property is there as you'd expect, with its getter and setter. But notice also the k__BackingField.
When we inspect it, it is comprised of the following code:
[CompilerGenerated]
[DebuggerBrowsable(DebuggerBrowsableState.Never)]
private string <Name>k__BackingField;
So we can see that there is definitely a private field in the background being generated for the property. We can also confirm that it actually uses that field by inspecting the get_Name getter on the Name property:
[CompilerGenerated]
{
return <Name>k__BackingField;
}
The getter for the Name property indeed returns the private field that was compiler generated for us.
I think what you're stuck on is why we would do this. Basically, if you simply have a public field on a class, you're giving permission for all-and-any other classes to set and get that field without any kind of checks or balances or rules about what goes on when setting or retrieving that value. For small applications that aren't very complex, this won't create an issue for you. But in the future when you're writing bigger applications or libraries, when you want to guarantee certain functionality behaves the same way all the time, this best practice will be necessary.