Yes, it does just save code. It's miles easier to read when you have loads of them. They're quicker to write and easier to maintain. Saving code is always a good goal.
You can set different scopes:
public string PropertyName { get; private set; }
So that the property can only be changed inside the class. This isn't really immutable as you can still access the private setter through reflection.
As of C#6 you can also create true readonly
properties - i.e. immutable properties that cannot be changed outside of the constructor:
public string PropertyName { get; }
public MyClass() { this.PropertyName = "whatever"; }
At compile time that will become:
readonly string pName;
public string PropertyName { get { return this.pName; } }
public MyClass() { this.pName = "whatever"; }
In immutable classes with a lot of members this saves a lot of excess code.