Does using the Goertzel algorithm actually give better frequency resolution?
Your understanding is correct.

Goertzel algorithms give almost exactly the same result as 1 bin of a DFT or FFT of the same length or number of samples (and where the FFT twiddle factors are generated by a trig recursion), when used for frequencies that are exactly integer periodic in the Goertzel length.  But many forms of the Goertzel algorithm provide only the magnitude, not the complex result or the phase of the FFT 1 bin result.  Also, computationally, a generic Goertzel may be numerically slightly less stable than a generic FFT.  For non-integer-periodic-in-aperture frequencies, the result is equivalent to a sinc interpolation between bins of a DFT or FFT of the same length (which can be slightly more accurate of an interpolation than the more typical parabolic interpolation of FFT results).

The interpolation might be said to increase resolution in a graphical sense (more plot points), or making it easier to visually spot the maxima, but not in the information theoretic sense, nor for better separating 2 close spaced spectral lines as 2 separate peaks.