POG’ll do octaves, the HOG will go between octaves and offer more routing options.

not quite.
an octave is twelve half-steps from your initial pitch.

the HOG works more based on the harmonics/overtones of your original signal-

let’s back up a step.
When you play a note on guitar, it’s not just a sine wave at, say, A440 [440Hz]; it’s also got overtones, probably at A880 [an octave above the original signal] and at E1319 [an octave and a fifth above the original signal], as well as other, quieter overtones.
By manipulating these and creating artificial ones, you can completely change the sound of the guitar into, say, a Hammond organ or a clavichord or basically anything a synthesizer would do.

The POG is more of a polyphonic digital pitch shifter, with options like ‘detune’ (chorus for the shifted pitches) and a low-pass filter (which can tame the fizzy stuff you can inadvertantly make with shifting things up in pitch).

The HOG offers control over attack and decay of the sounds created [the volume slopes at the start and the end of a sound – for instance, a hi-hat has both a fast attack and decay, an orchestra string section swell has a slow attack and decay, and a cymbal has a fast attack and slow decay], as well as several other controls/routing options like glissandro [think a slide on a fretless bass], filtering, wah-esque filter sounds, etc. Also, with a HOG, you can use an expression pedal (think like a wah pedal but able to control more than just the things that make it go ‘squonk’.)

if you’re looking for something that just drops your pitch by a half-step, the Ring Thing would be the way to go, or the Morpheus drop tune.