Difference between Gentzen and Hilbert Calculi What is the difference between Gentzen and Hilbert Calculi?
From  my understanding of Rautenberg's Concise Introduction to Mathematical Logic, Gentzen calculus is based on sequents and Hilbert calculus, on tautologies.
But isn't every Gentzen sequent a tautological modus ponens?
For instance, the sequent $X\vdash a \wedge b |X \vdash a,b$ can be written as the tautology $\forall X \forall a \forall b\:( X\vdash a \wedge b \rightarrow X \vdash a,b$), can't it?

You can see a very detailed overview into : Francis Pelletier & Allen Hazen, Natural Deduction :

Sequent Calculus was invented by Gerhard Gentzen (1934), who used it as
  a stepping-stone in his characterization of natural deduction [...]. It is a very general characterization of a proof; the basic notation being
  $ϕ_1,\ldots,ϕ_n ⊢ ψ_1,\ldots,ψ_m$, which means that it is a consequence of the premises $ϕ_1,\ldots,ϕ_n$ that at least one of $ψ_1,\ldots,ψ_m$ holds.
If $\Gamma$ and $\Sigma$ are sets of formulas, then $\Gamma ⊢ \Sigma$ means
  that it is a consequence of all the formulas of $\Gamma$ that at least one of the formulas in $\Sigma$ holds.
Sequent systems take basic sequents such as $ϕ ⊢ ϕ$ as axiomatic, and then a set of rules that allow one to modify proofs, or combine proofs. The modification rules are normally stated in pairs, ‘$x$ on the left’ and ‘$x$ on the right’: how to do something to the premise set $\Gamma$ and how to do it to the conclusion set $\Sigma$. So we can understand the rules as saying “if there is a consequence of such-and-so form, then there is also a consequence of thus-and-so form”.
These rules can be seen as being of two types: structural rules that characterize the notion of a proof, and logical rules that characterize the behavior of connectives. For example, the rule that from $\Gamma ⊢ \Sigma$ one can infer $\Gamma,ϕ ⊢ \Sigma$ (“thinning on left”) characterizes the notion of a proof (in classical logic), while the rule that from $\Gamma,ϕ ⊢ \Sigma$ one can infer $\Gamma, (ϕ∧ψ) ⊢ \Sigma$ (“∧-Introduction on left”) characterizes (part of) the behavior of the logical connective ∧ when it is a premise. 

In the paper you can find an historical overview and several useful discussions; relevant for Rautenberg's textbook, see :


*

*3.4 From Sequent Natural Deduction to Sequent Calculus

*3.8 Natural Deduction with Sequences .

