Difference of Beilinson conjecture and equivariant Tamagawa number conjecture 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.
 A: 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.
