Mostly I refer you to my answer here and also this question.
To answer the question about (co)fibrations: No, there is no notion corresponding to (co)fibration in the (∞,1)-category associated to a model category. After all, being a (co)fibration has no homotopical information: every map is equivalent to a (co)fibration. For the sorts of things you need the (co)fibrations to define in model categories, such as homotopy (co)limits, you can give direct definitions in terms of mapping spaces in the (∞,1)-category.
There are two sensible notions of "sameness" of model categories: categorical equivalence, by which I mean an equivalence of categories which preserves each of the three classes of arrows, and Quillen equivalence. This is a lot like the difference between two objects in a model category being isomorphic or merely weakly equivalent, though I don't think anyone has a framework in which to make this idea precise. When you consider, say, the projective and injective model structures on a diagram category, these model structures are Quillen equivalent but not categorically equivalent. They have different 1-categorical properties (it's easy to describe left Qullen functors out of the projective model structure and left Quillen functors into the injective model structure) but they model the same homotopy theory. The passage to associated (∞,1)-categories eliminates the distinction between categorical equivalence and Quillen equivalence: two model categories are Quillen equivalent if and only if their associated (∞,1)-categories are equivalent. (Actually, I am not sure whether there are some technical conditions needed for the last assertion, but if so they are satisfied in practice.)
As a preface, I think that this question should be viewed as analogous to "what are the advantages of ZFC over type theory" or vice versa. We're talking about foundations -- in principle, it doesn't matter what foundations you use; you end up with an equivalent model-independent theory of $\infty$-categories.
The paradigm is "use simplicial categories for examples; use quasicategories for general theorems". There are a lot of constructions which are simpler in quasicategories. I tend to think that a lot of the difference is visible at the model category level: the Joyal model structure on $sSet$ is just much nicer to work with than the Bergner model structure on $sCat$ (of course, the latter is still theoretically very important, at the very least for the purposes of importing examples which start life as simplicial categories). Some of these differences are:
The Joyal model structure is defined on a presheaf category.
The Joyal model structure is cartesian, making it much easier to talk about functor categories.
In the Joyal model structure, every object is cofibrant.
There's a synergy between (1) and (2) -- if $X$ is a quasicategory and $A$ is any simplicial set, then the mapping simplicial set $Map(A,X)$ gives a correct model for the functor category from $A$ to $X$ -- you don't need to do any kind of cofibrant replacement of $A$. This is nicely explained in Justin Hilburn's answer.
(2) is quite convenient. For example, in general frameworks like Riehl and Verity's $\infty$-cosmoi, a lot of headaches are avoided by assuming something like (2).
Here are a few examples of some things which are easier in quasicategories -- I'd be curious to hear other examples folks might mention!
The join functor is very nice.
Consequently (in combination with the nice mapping spaces), limits and colimits can be defined pretty cleanly.
An example of a theorem proven in HTT using quasicategories which I imagine would be hard to prove (maybe even to formulate) directly in simplicial categories is the theorem that an $\infty$-category with products and pullbacks has all limits. The proof uses the fact that the nerve of the poset $\omega$ is equivalent to a 1-skeletal (non-fibrant) simplicial set, and relies on knowing how to compute co/limits indexed by non-fibrant simplicial sets like this.
- The theory of cofinality is very nice, arising from the (left adodyne, left fibration) weak factorization system on the underlying category -- I imagine it would be quite complicated with simplicial categories.
Roughly at this point in the theory, though, one starts to have enough categorical infrastructure available that it becomes more possible to think "model-independently", and the differences start to matter less.
Here's a few more:
When you take the maximal sub-$\infty$-groupoid of a quasicategory, it is literally a Kan complex, ready and waiting for you to do simpicial homotopy with. This is especially nice when you take the maximal sub-$\infty$-groupoid of a mapping object -- which doesn't quite make the model structure simplicial, but it's kind of "close".
The theory of fibrations is pretty nice in quasicategories -- just like in ordinary categories, left, right, cartesian, and cocartesian fibrations are "slightly-too-strict" notions which are very useful and have nice properties like literally being stable under pullback. I don't know what the theory of such fibrations looks like in simplicial categories.
The fact that $sSet$ is locally cartesian closed is sneakily useful. Even though $Cat_\infty$ is not locally cartesian closed, there's a pretty good supply of exponentiable functors, and it's not uncommon to define various quasicategories using the right adjoint to pullback of simplicial sets. (Rule of thumb: in HTT, when Lurie starts describing a simplicial set by describing its maps in from simplices over a base, 90% of the time he's secretly describing the local internal hom of simplicial sets.)
Best Answer
It seems that the first question only makes sense for marked simplicial sets $X$ over $S$ where every edge of $X$ is marked (otherwise, the slice category is not equivalent to marked simplicial sets over $X$). Under this assumption, the answer is yes at least if $X$ is fibrant (so that the underlying map of simplicial sets $X \rightarrow S$ is a right fibration).
If you take any model category ${\mathbf A}$ for higher category theory and take the slice category ${\mathbf A}_{/X}$ for some fibrant object $X$, it will be a model for higher categories $Y$ with an arbitrary functor $p: Y \rightarrow X$. If you want to enforce the requirement that $p$ should be a Grothendieck fibration, you need to modify the definitions somehow. In the quasicategory model, this is what the markings are for.