In the Development with the Force.com Platform book on pg 130, they mention that an Array's length is fixed when you initialize it and that multidimensional arrays are not supported. I just tried using the .add function on an array and it worked so I'm sure what they are talking about, but I couldn't make an array of arrays (which could have just been because I didn't have the right syntax). Ignoring syntax errors, there seems to be some sort of difference and thus maybe an (even if just slight) advantage to using one over the other?
There really aren't arrays in Apex, but you can use the array notation to declare lists. From the documentation on lists:
When using one-dimensional lists of primitives or sObjects, you can also use more traditional array notation to declare and reference list elements. For example, you can declare a one-dimensional list of primitives or sObjects by following the data or sObject type name with the  characters
To answer the question about fixed arrays (lists), it all depends on how you try to put elements in them:
Even though the size of the previous String array is defined as one element (the number between the brackets in new String1), lists are elastic and can grow as needed provided that you use the List add method to add new elements. For example, you can add two or more elements to the colors list. But if you’re using square brackets to add an element to a list, the list behaves like an array and isn’t elastic, that is, you won’t be allowed to add more elements than the declared array size.
In short you can use the array notation to declare a list. All list methods can be used with it. More
The main difference is that Lists can be multi dimensional. i.e, list of lists.