<p>Why do we have two standard filament sizes, 1.75&nbsp;mm and 3&nbsp;mm? Does it really make a difference when printing? Or is the 1.75&nbsp;mm just for smaller printers?</p>

<p>In what situations should I be using 1.75&nbsp;mm?</p>

<p>When should I be using 3&nbsp;mm?</p>

There's no appreciable difference. Just use the filament that fits your particular printer.
If you don't yet have a printer, then I'd get one that uses 1.75 mm filament:
1.75 mm is increasingly becoming the "standard", thus being easier to get. Some filaments are not available as 3 mm.
1.75 mm filament allows for finer control, because feeding in 1 mm of filament corresponds to less plastic extruded.
1.75 mm filament requires less force to extrude. Compressing 1.75 mm down to 0.3 mm takes less force than doing the same to 3 mm filament.
However, the advantages are fairly minor. I don't see any reason to replace a functioning 3 mm extruder with a 1.75 mm one (yet).