I would not add much by saying that "Introduction to MetaMathematics" (IM) remains a masterpiece, even though the style is a bit oldish...
On the the other hand, "Mathematical Logic" (ML) brings a definite plus, but is by no means a replacement, rather a necessary complement.
As I planned to study both, the problem posed was the order in which one should approach those books : Historically ? By increasing or decreasing difficulty ? In parallel, in order to see how Kleene's ideas -- and the field -- have evolved between 1952 and 1966, and subject by subject ?
I chose the third an most difficult path... And the journey was a thrill !
Here is how I planned this strange exploration : IM, ch. 1 to 7 ; ML, ch. 1 to 4 ; IM, ch. 8 ; IM, Part III ; ML, ch. 5 : IM, ch. 14 ; ML, ch. 6 ; IM, ch. 15.
ML is certainly less difficult but contains a fair amount of footnotes linking it to IM, i.e. studying IM is simply inevitable and enjoyable, even though some parts are really tough and must be "examined in a cursory manner", as suggested by Kleene, e.g. ch. 14 & 15.
IM, part III, is a thorough treatment of recursive functions, the best in my opinion and is not part of ML.
All in all, the two together rank very high in logic books, perhaps highest.
This book now stands in my list of outstanding books on logic :
1. A. Tarski's "Introduction to Logic", a jewel, followed by P. Smith's superb entry-point "An introduction to Formal logic" and the lovely "Logic, a very short introduction" by Graham Priest
2. D. Goldrei's "Propositional and Predicate calculus"
3. Wilfrid Hodges' "Logic", followed by Smullyan's "First-order logic".
4. P. Smith's "An introduction to Gödel's theorems".
5. Kleene's "Introduction to metamathematics" & "Mathematical Logic".
6. G. Priest's " Introduction to non-classical logic".
Hence forgetting altogether Van Dalen's indigestible "Logic & Stucture" as well as the even more indigestible Enderton, Mendelson & al...