As stated in the title, I am wondering the main difference between Beilinson conjecture and eTNC. If I read correctly, I can see that there are many literature treating both conjectures in the same line - special values of L-functions. If my impression is completely wrong, I am sorry for this stupid question. But, if they have some commonness, it would be greatly helpful if anyone can clarify their difference.



Difference of Beilinson conjecture and equivariant Tamagawa number conjecture
The key difference between these conjectures is the coefficient ring that is involved. You also are leaving out an important "middle" conjecture -- the (non-equivariant) Tamagawa number conjecture, as formulated in Bloch and Kato's article in the Grothendieck Festschrift -- and knowing what this conjecture says might clarify the relation between the other two conjectures a bit.


In Beilinson's conjecture, the coefficient ring is $\mathbf{Q}$: the conjecture predicts the leading terms of $L$-functions up to rational factors, in terms of the determinant of a regulator matrix defined using a $\mathbf{Q}$-basis of some motivic cohomology space.


In the Bloch--Kato Tamagawa number conjecture (TNC), the coefficient ring is $\mathbf{Z}\_\ell$ for some prime $\ell$, and the conjecture predicts the leading terms of $L$-functions up to $\ell$-adic units. The extra integral structure comes from comparing motivic cohomology with Galois cohomology, which (unlike motivic cohomology) works well with $\mathbf{Z}\_\ell$ coefficients. 


In the equivariant Tamagawa number conjecture (ETNC), one introduces further extra structure by considering a Galois extension $K / k$ and studying the arithmetic of a motive over $k$ base-extended to $K$, which means that all the cohomology groups are modules over the group ring $\mathbf{Z}\_\ell[G]$ where $G = \operatorname{Gal}(K/k)$ is some finite group. Then the conjecture relates the leading terms of equivariant (i.e. group-ring-valued) $L$-functions to the isomorphism classes of cohomology groups as $\mathbf{Z}\_\ell[G]$-modules.


(You could also formulate an "equivariant Beilinson conjecture" involving $\mathbf{Q}[G]$-modules, but it would be fairly trivally equivalent to the original Beilinson conjecture for each of the motives $M \otimes \rho$ where $\rho$ varies over representations of $G$. For the same reason, ETNC reduces to TNC when $\ell \nmid \#G$. The interesting cases are when $G$ is an $\ell$-group, in which case there are non-trivial congruences between $\mathbf{Z}\_\ell$-representations of $G$ which the ETNC detects.)


So, to sum up, the triple of conjectures (Beilinson conj, TNC, ETNC) all seek to relate $L$-values to arithmetically-interesting cohomology modules, but the cohomology modules are over the rings $(\mathbf{Q},\ \mathbf{Z}\_\ell,\ \mathbf{Z}\_\ell[G])$ respectively.


