Property with getter only vs. with getter and private setter

public string MyProp { get; } - This is introduced in C# 6.0. And such properties are called read-only auto-properties. Assignments to such members can only occur as part of the declaration or in a constructor in the same class. You can read detailed explanation about it in that MSDN article or in Jon Skeet blog. As explained in that article, such property solves four problem automatically:

• A read-only-defined backing field
• Initialization of the backing field from within the constructor
• Explicit implementation of the property (rather than using an auto-property)
• An explicit getter implementation that returns the backing field

public string MyProp { get; private set; } - This means that the property is read-only in the outside of this class, but you can change it's value inside of this class.

By the way, you can set read-only auto-properties value using new auto-initialize syntax which is again introduced in C# 6.0:

public string MyProp { get; } = "You cannot change me";

It is equal to this code for the previous versions of C#:

private readonly string myProp = "You cannot change me"
public string MyProp { get { return myProp ; } }

Or, this in C# 6.0:

public string MyProp { get; }
protected MyClass(string myProp, ...)
{
    this.MyProp = myProp;
    ...
}

is equal to this in the previous versions:

private readonly string myProp;
public string MyProp { get { return myProp; } }
protected MyClass(string myProp, ...)
{
    this.myProp = myProp;
    ...
}

In the first one no-one can set the properties value, in the second case at least the class itself can change it. Having said this only the first one is a real "read-only"-property.

However consider combining it with a readonly-backend-field:

private readonly string field;
public string MyProp { get { return this.field; } }

Now any attemp to change the value of either the property itself or the backing-field will fail. This is identical to the syntax introduced in C#6 where the baking-field is automatically added to be reradonly:

public string MyProp { get; }

However as you already assumed for other classes both properties work same, meaning are not ment to be modified in any way.