Immutability is important in multi-threaded programs, because then you know that one thread won't corrupt a value used in another thread. But it's also useful in a single-threaded program.
Here's a simple example:
Integer i=Integer.valueOf(17);
foo(i);
bar(i);
You might well want to know, What value is passed to bar()?
Suppose that foo() is a big, complex function. In this example, I know for an absolute fact that when foo completes, i is still equal to 17, because an Integer is immutable. Were that not true, I would have to study foo to tell if it might be changed or not.
Here's a slightly more complex example. Suppose I have some object that resembles an Integer, but is mutable. Let's call it MutableInteger. Then say I write this:
MutableInteger currentInventory=findQtyInInventory();
MutableInteger neededInventory=currentInventory; // copy current for starters
... bunch of other code ...
neededInventory.subtract(allocatedToSales);
currentInventory.add(arriving);
... bunch of more code ...
if (neededInvenory.compareTo(currentInventory)>0)
display("Shortage!");
Do you see the problem with the above? neededInventory and currentInventory point to the same object. All the adds and subtracts are really acting on the same value, not two different values, so when we get to the test, it will always be equal. The above code would never work if the objects are mutable. If they are immutable, the adds and subtracts would have to return a result object rather than updating in place, and that would work.
Years ago I used a Fortran compiler where integers WERE mutable. We had a function that accepted several parameters, one of them an integer. In some rare cases, the function updated the integer. Then one day someone wrote a call to this function passing the constant "2" as the integer. The function decided to update the parameter, thus changing the "constant" 2 to 1! Every other place in the program that used a constant 2 now mysteriously got the value 1 instead. This took a long time to debug.