When to use 1.75 mm vs 3 mm filament?
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&nbsp;mm filament:


1.75&nbsp;mm is increasingly becoming the "standard", thus being easier to get. Some filaments are not available as 3&nbsp;mm.
1.75&nbsp;mm filament allows for finer control, because feeding in 1&nbsp;mm of filament corresponds to less plastic extruded.
1.75&nbsp;mm filament requires less force to extrude. Compressing 1.75&nbsp;mm down to 0.3&nbsp;mm takes less force than doing the same to 3&nbsp;mm filament.


However, the advantages are fairly minor. I don't see any reason to replace a functioning 3&nbsp;mm extruder with a 1.75&nbsp;mm one (yet).