Relation between Type Assignment system (TA) and Hindley-Milner system
System F and its subsystem HM have a type former for universal quantification:

$$
   \tau \quad::=\quad \forall x.\tau \ |\ ...
$$

which the system in Hindley/Seldin doesn't have. That is the key difference.

Now System F doesn't have decidable type-inference, and HM is a way of combining type-inference with reasonably expressive parametric polymorphism. HM achieves this by allowing only outermost universal quantification, i.e. all types are of the form

$$
   \forall x_1 \forall x_2 ... \forall x_n.\tau
$$

where $\tau$ is quantifier free (and $n \geq 0$). HM gives a rule system that ensures that only programs that can be typed in this way are admissible. This is achieved by "let-polymorphism". The system in Hindley/Seldin doesn't do any of that. Later, in Chapter 13, Hindley/Seldin introduce pure type systems (PTS), of which System F is a special case.  I'm not sure if HM can be expressed as a PTS.



The question of strong normalisation is orthogonal. System F and HM are strongly normalising, but that can easily be remedied by introducing fix-point combinators or similar. The paper  Principal type-schemes for functional programs  by L. Damas and R. Milner even states this:
"For example, recursion is omitted since it can be introduced by simply adding
    the polymorphic fixed-point operator ..." The introduction of fixpoints, making the system Turing complete, poses no issues from the point of view of type inference.