It appears that the two algorithms may have chosen a different branch of complex log.
Table[x=RandomReal[{-10,10}];
N[-Log[(x^2-2x+2)Sqrt[x^4-2x^3+3x^2+2x+1]+x^4-3x^3+5x^2-2x]-
Log[-2x+5x^2-3x^3+x^4+(-2+2x-x^2)Sqrt[1+2x+3x^2-2x^3+x^4]]],{20}]
which displays
{-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 ],
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I,
-1.38629-3.14159 I}
With a bit of detective work you should be able to determine the exact value that is approximately -1.38629.
If you try to plot the real or imaginary component of that difference over a larger range, perhaps over -100..100 or more, then you may see what looks like noise for the larger values of x. But if you use increased precision, perhaps Plot[Re[N[difference,64]],{x,-100,100},WorkingPrecision->64]
then the noise is removed and the difference between the two solutions again appears to be constant.
The difference between the two results might be -2*Log[2]-Pi*I