Let’s give a short answer first. 0V is a point in the circuit that you assign as your zero volt potential and it is usually referred to as “common ground”. 0 ground is a 0V point literally connected to external grounding facility.
To explain the differences in more depth we need to mention three terms: 0V potential, common junction and ground.
If you use a 9V battery in your new circuit you have a choice to give one of the leads an arbitrary potential. You can call it 1000V if that’s what you like because only potential differences matter. If you assign the negative lead potential of 0V it means that the positive lead is now +9V but you could instead assign the positive lead 0V, which makes the negative lead -9V. It is thus common to pick a point and call it 0V, usually we like a point where most components are connected thus we call it “the common junction”. Sometimes we call it “circuit ground” but it is not necessarily true ground.
Independently, you have a choice to connect one junction of your circuit to external grounding. This sounding is usually connected eventually to the home safety ground lead that is kept at the same potential as the ground around the building via conductive leads buried in the ground or connected to ground-exposed metal plumbing. This is real ground, also known as the safety ground. There are even different symbols for each ground (see below).
Usually the common junction of a circuit is connected to the external ground but sometimes we pick another junction for connecting to external ground. grounding becomes important when multiple pieces of equipment are connected to each other, such as the cabe TV cox shield between the CATV company and your cable box. This is so ubiquitous that in the practice of circuit design the 0V junction is called common even when it is not connected to any external ground, as happens in the circuits of a wireless telephone.