In the former, any code included is executed as soon as class B is created. In the latter, however, until you actually use the object, it won't be instantiated.
You can see the difference here:
class A { println("Creating a new A") }
class B {
val a1 = new A { println("a1"); def foo = 1 }
object a2 extends A { println("a2"); def foo = 1 }
}
scala> val b = new B
Creating a new A
a1
b: B = B@1176e8a
scala> b.a2.foo
Creating a new A
a2
res0: Int = 1
There are also hidden differences in what the created .class files are named and such; and of course the two have different types.