roam vs gank
Who moves. A gank is done by the jungler, whose job it is; a roam is done by a laner, who leaves their lane for it. A roam is a gank that costs minions.
roam
Roaming is leaving your lane to act elsewhere: gank another lane, place vision, help on an objective. It is the mid's or the support's move, and it has a price: the wave you leave behind, the gold and experience lost. "pushing wave dont roam" is the reminder of that price.
open the page →gank
A gank is the surprise attack on a lane. The word does not say who does it, but in practice it is almost always the jungler, because they lose nothing by moving: the whole map is their lane. "gank mid", "gank now".
open the page →Why they get mixed up
A laner who roams often ends up ganking, hence the mix-up. The distinction matters to whoever receives the call: "mid roaming" is an alert for the other lanes, "jgl ganking bot" is information about where the jungler is. And to whoever announces it: telling your team "roaming" is warning them you are abandoning your wave.