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2 added 86 characters in body

[I'm a novice, and this got posted out of order: it answers Bak's question below.]

Sure, I can provide that. The cited reference was published in 1995, which was well before details of symmetric or orthogonal spectra were available, so it gives a fair amount of background but only refers to EKMM spectra for a modern category. There is a paper (Mandell, May, Schwede, Shipley) that compares all choices except EKMM, and there are various papers that compare those choices with EKMM, starting with a paper by Schwede. Those papers are maybe more technical than you want. A recent survey paper compares the various approaches philosophically: see Sections 11 and 12 of my paper

What precisely are $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra? Geometry \& Topology Monographs 16(2009), 215--282.

That gives references and is fairly independent of Sections 1-10. It starts with a theorem (11.1) of Gaunce Lewis explaining that there is no ideal choice of category: if you assume your category has all the good properties you want, you reach a contradiction. The incompatibility comes when you ask for a homotopically meaningful symmetric monoidal structure on your category of spectra that also has a homotopically meaningful monoidal adjunction $(\Sigma^{\infty},\Omega^{\infty})$ relating spaces and spectra. I'm old-fashioned maybe, but I think spaces are still kind of important.

EKMM comes as close as possible to having such an adjunction, with the related advantage that all objects are fibrant and the related disadvantage that the sphere spectrum is not cofibrant. Symmetric and orthogonal spectra have the advantage that they are significantly easier to define and the sphere spectrum is cofibrant.
The simplicial version of symmetric spectra has the advantage that it is especially well-suited to adaptation to the motivic world. Orthogonal spectra have the advantage that they are much better suited for equivariant and parametrized generalizations than symmetric spectra. Common features are captured by the web of Quillen equivalences relating not just all known constructions but all possible good'' model categories of spectra: there is an axiomatization, due to Shipley; symmetric spectra play a privileged role in the proof.

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Sure, I can provide that. The cited reference was published in 1995, which was well before details of symmetric or orthogonal spectra were available, so it gives a fair amount of background but only refers to EKMM spectra for a modern category. There is a paper (Mandell, May, Schwede, Shipley) that compares all choices except EKMM, and there are various papers that compare those choices with EKMM, starting with a paper by Schwede. Those papers are maybe more technical than you want. A recent survey paper compares the various approaches philosophically: see Sections 11 and 12 of my paper

What precisely are $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra? Geometry \& Topology Monographs 16(2009), 215--282.

That gives references and is fairly independent of Sections 1-10. It starts with a theorem (11.1) of Gaunce Lewis explaining that there is no ideal choice of category: if you assume your category has all the good properties you want, you reach a contradiction. The incompatibility comes when you ask for a homotopically meaningful symmetric monoidal structure on your category of spectra that also has a homotopically meaningful monoidal adjunction $(\Sigma^{\infty},\Omega^{\infty})$ relating spaces and spectra. I'm old-fashioned maybe, but I think spaces are still kind of important.

EKMM comes as close as possible to having such an adjunction, with the related advantage that all objects are fibrant and the related disadvantage that the sphere spectrum is not cofibrant. Symmetric and orthogonal spectra have the advantage that they are significantly easier to define and the sphere spectrum is cofibrant.
The simplicial version of symmetric spectra has the advantage that it is especially well-suited to adaptation to the motivic world. Orthogonal spectra have the advantage that they are much better suited for equivariant and parametrized generalizations than symmetric spectra. Common features are captured by the web of Quillen equivalences relating not just all known constructions but all possible good'' model categories of spectra: there is an axiomatization, due to Shipley; symmetric spectra play a privileged role in the proof.