These are delicate issues.
Both $C^*$-algebras and $W^*$-algebras (also known as von Neumann algebras) are used to describe the observables of a quantum system. Properly speaking, observables
are self-adjoint elements of the algebra.
$W^*$ algebras are relevant in physics since they are isomorphic to concrete $C^*$-algebras of bounded operators in complex Hilbert spaces with the further property that they are closed with respect to weak and strong operator topology and satisfy the so-called double commutant theorem.
Finally they are generated (in the complex case) by their set of orthogonal projectors interpretable as the set of elementary observables of the quantum system, providing this way a natural interpretation of the spectral theorem and a justification of the correspondence
observable - selfadjoint operator.
The closure of von Neumann algebras with respect to the strong operator topology is the feature that makes them more interesting than simple $C^*$-algebra of operators. Simultaneously it makes them very rigid structures not suitable to describe some quantum systems in general situations (e.g., locally covariant QFT in curved spacetime).
I mention two consequences.
(a) If a (bounded) self-adjoint operator belong to a $C^*$-algebra of operators representing the observables of a quantum system, in general, its projection-valued measure does not belong to the algebra -- it would happen if the algebra were von Neumann -- because it is obtained from the algebra with operations which are not uniformly continuous but just strongly continuous.
(b) Similarly if a symmetry is described by a strongly continuous unitary group of operators of a $C^*$-algebra of concrete operators, generally speaking, the projection-valued measure of the generator of the group does not belong to the $C^*$-algebra. It happens if the algebra is von Neumann instead.
Also superselection rules enter the theory differently depending on the choice between $C^*$-algebras and von Neumann algebras to describe the observables of the system.
The use of non-normal states (of either $C^*$ or $W^*$ algebra) is sometimes compulsory, since there are mathematical results forcing two irreducible representations of the same algebra to be non-unitarily equivalent and these representations are constructed upon different algebraic states through the GNS construction. As a consequence the said (pure) states cannot belong to the same folium. Haag's theorem is the most known result of this sort. However there are other cases in QFT and in statistical mechanics discussing extended systems and spontaneously broken symmetry.
References. There are many papers on these subjects, but usually they are too mathematically minded or mostly directed towards QFT.
Haag's book: Local Quantum Physics (Second Revised and Enlarged Edition).
Springer, Berlin (1996) mentions several general results though the book is evidently interested in QFT.
Emch's book: Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley-Interscience, New York (1972) is another (maybe a bit hold) source. There also Jordan algebras are exploited as crucial tools.
Strocchi's introductive textbook: An Introduction To The Mathematical Structure Of Quantum Mechanics: A Short Course For Mathematicians. World Scientific, Singapore (2005). This book is directly related to quantum mechanics but it is also its limit.
A classic reference written as an encyclopedia is Beltrametti-Cassinelli's book Beltrametti, E.G., Cassinelli, G.: The logic of quantum mechanics. Encyclopedia of Mathematics and its Applications, vol. 15, Addison-Wesley, Reading, Mass.,(1981).
A modern treatise where several problems related to those you raised (very complete but perhaps sometimes written into a too concise fashion)
Blank, J., Exner, P., Havlicek, M.: Hilbert Space Operators in Quantum Physics, second edition, Springer, Berlin (2007).
A bit discussion on all these issues appears in my lectures of some years ago
A much more extended discussion -- including a list of axioms for QM -- can be found in my 2013 book http://www.springer.com/it/book/9788847028340 (I am sure that the file of the book stays in library genesis!)
and especially in the 2nd corrected and enlarged edition
http://www.springer.com/it/book/9783319707051 in print, where I have added several sections on these subjects.
A recent book containing chapters of several authors (I wrote Ch.5 together with I.Khavkine) about applications of algebraic formalism to QFT is
Brunetti R., Dappiaggi, C., Fredenhagen, K., and Yngvason, J. (Eds): Advances
in Algebraic Quantum Field Theory, Springer-Verlag, Berlin, (2015).