Comparing a unimodular transformation to an arbitary rotation matrix was trivial.  To do the same for an Euler angle rotation matrix and arbitrary matrix is less so because of sign differences.  Here is a dumb symbol expansion of the full rotation on a general vector in Pauli matrix form to have a look at the two results and see if there is anything striking.  Ans.: no, there is not.  Both expressions are messy even after FullSimplify.  Was able to extract some structure from the Euler rotation method by avoiding full expansion, and then factoring out terms of x,y,z.  However, really want to factor out the vector that is uneffected by the rotation (to determine the normal direction), and then figure out the rotation applied to the remainder.

ClearAll[sigmaX, sigmaY, sigmaZ, sigma, sigmaDot, v, d, di, sigmaDotNcap, ncap, r, r1i, r1 ]
sigmaX = {{0, 1}, {1, 0}} ;
sigmaY = {{0, -I}, {I, 0}} ;
sigmaZ = {{1, 0}, {0, -1}} ;
sigma = {sigmaX, sigmaY, sigmaZ} ;

sigmaDot[x_, y_, z_] = sigmaX x + sigmaY y + sigmaZ z ;
v = sigmaDot[x,y,z] ;

dOneHalf[alpha_, beta_, gamma_] = {{Exp[-I(alpha+gamma)/2] Cos[beta/2], -Exp[-I(alpha-gamma)/2] Sin[beta/2]},
{Exp[I(alpha-gamma)/2] Sin[beta/2],Exp[I(alpha+gamma)/2] Cos[beta/2]}} ;
d = dOneHalf[α, β, γ] ;
di = Inverse[dOneHalf[α, β, γ]] // Simplify ;

(*di . v . d // Simplify // Factor ;*)

(* rotation of theta about ncap = (sin\alpha cos\beta, sin\alpha \sin\beta, cos\alpha) : e^{-i sigma . ncap theta/2} *)
ncap[alpha_, beta_] = {Sin[alpha] Cos[beta], Sin[alpha] Sin[beta], Cos[alpha]} ;
sigmaDotNcap[alpha_, beta_] = Sum[sigma[[i]]  ncap[alpha, beta][[i]], {i, 3}] // Simplify ;
(*sigmaDotNcap[α, β] // TraditionalForm*)
r[theta_, alpha_, beta_] = Cos[theta/2] DiagonalMatrix[{1,1}] -I Sin[theta/2] sigmaDotNcap[alpha, beta] ;
(*r[θ, α, β] // TraditionalForm*)

r1 = r[θ, α, β] // Simplify ;
r1i = Inverse[r[θ, α, β] ] // Simplify ;
(*r1 // TraditionalForm
r1i // TraditionalForm*)

r1 // TraditionalForm
rr = r1i . v . r1 // Simplify ;
Grid[{
{"r' = ", "x \!\(\*SubscriptBox[\(m\), \(\(x\)\(\\ \)\)]\)+ y \!\(\*SubscriptBox[\(m\), \(\(y\)\(\\ \)\)]\)+ z \!\(\*SubscriptBox[\(m\), \(\(z\)\(\\ \)\)]\)"},
{"(Subscript[m, x]) = ",
Coefficient[rr,x,1] // TraditionalForm},
{"(Subscript[m, y]) = ",
Coefficient[rr,y,1] // TraditionalForm},
{"(Subscript[m, z]) = ",
Coefficient[rr,z,1] // TraditionalForm}}]

With a = e^i\alpha/2, b = e^i\gamma/2, c = cos\beta/2, s = sin\beta/2.  Any simpler?  Not much.  With just a = e^i\alpha/2, b = e^i\gamma/2 things are a bit simpler.  In particular, the z contributions stand out.

ClearAll[df, d2, d2i]
df[a_, b_, c_, s_] = {{c/(a b), -s b/a},{s a/b, c a b }} ;
d2 = df[a,b,c, s] ;
d2i = Inverse[d2] /. ( c^2 + s^2 -> 1) /. c -> Cos[β/2] /. s -> Sin[β/2] ;
d2 = d2 /. ( c^2 + s^2 -> 1) /. c -> Cos[β/2] /. s -> Sin[β/2]  ;
d2 // MatrixForm
d2i // MatrixForm

rv = d2i . v . d2 //  FullSimplify ;
rv // TraditionalForm
(*((rv - z sigmaZ Cos[β] // Expand) /. Power[a,p_] :>  E^(Iα p/2) /. Power[b,p_] :>  E^(I β p/2)) // Simplify // TraditionalForm*)
(*/. Power[a,2] :>  E^(Iα) /. Power[a,-2] :>  E^(-Iα)*)
(*/. a^2 -> E^(Iα) /. b^2 -> E^(Iγ) // MatrixForm*)
(*/. c^2 -> (1 + Cos[β])/2 /. s^2 -> (1 - Sin[β])/2 // MatrixForm*)

Grid[{
{"r' = ", "x \!\(\*SubscriptBox[\(m\), \(\(x\)\(\\ \)\)]\)+ y \!\(\*SubscriptBox[\(m\), \(\(y\)\(\\ \)\)]\)+ z \!\(\*SubscriptBox[\(m\), \(\(z\)\(\\ \)\)]\)"},
{"(Subscript[m, x]) = ",
Coefficient[rv,x,1] // TraditionalForm},
{"(Subscript[m, y]) = ",
Coefficient[rv,y,1] // TraditionalForm},
{"(Subscript[m, z]) = ",
Coefficient[rv,z,1] // TraditionalForm}}]

Used above: How to factor polynomial with matrix coefficients: http://mathematica.stackexchange.com/a/7674/10

ClearAll[x, y, z, a1, a2, a3,a4, b1, b2, b3, b4, c1, c2, c3, c4]
m = {{x a1 + y b1 + z c1, x a2 + y b2 + z c2},{x a3 + y b3 + z c3, x a4 + y b4 + z c4}} ;
m // TraditionalForm
Coefficient[m,x,1] // TraditionalForm
Coefficient[m,y,1] // TraditionalForm
Coefficient[m,z,1] // TraditionalForm