Power Spectral Density (PSD) from autocorrelation
I am not sure I follow your exact questions.


  Is there a specific mathematical reason as to why the latter method
  appears to carry more energy at the higher frequencies?


No. The two methods are equivalent and will match to the numerical precision you are using.


  Auto-correlation function is analogous to a low-pass filter, meaning
  low frequencies are getting through, and high frequencies are
  attenuated out (gradual loss) at a certain cutoff frequency.


I do not understand this claim. How is auto-correlation a low-pass filter? Auto-Correlation is auto-correlation.


  Technically both methods shows some form of aliasing. So are there
  different kinds of aliasing? One is observed to aliased over a range
  of wavenumbers, while the other shows a spike.


Your input signal is already sampled and you are not changing the sampling rate, so I do not understand this either.

If the PSDs as estimated from two methods do not match, then there is a mistake in your code. Below is an example using octave/matlab code:

N = 1000;
x = rand(N,1);
Nconv = 2*N - 1;

% PSD from autocorrelation.
a = conv(x,flipud(conj(x)));
A = abs( fftshift(fft(a)) );

% PSD from FFT.
X = fftshift(fft(x,Nconv));
A2 = X .* conj(X);

% The residual between the two estimates.
err = 10*log10( norm(A - A2) / norm(A) );