Well, the first way to distinguish them is that a hive mind is a mind. So anything mindless is clearly not a hive mind. For example, ants are eusocial insects, but as far as we know, have no mind, whether hive or other.
Let's keep with the ants and imagine an advanced ant species that evolves a mind. This can happen in two ways: On one hand, every ant by itself might develop a mind. Then this would not be a hive mind; those insects would be eusocial organisms with an individual mind. Each insect would experience itself as an individual entity. If you hurt an ant, only that ant would feel the pain. An eusocial species with individual minds would likely form something like a fascist society: There are individuals, but they only matter as part of the collective.
On the other hand, one could imagine that the ant colony as a whole would develop a mind, just as our body as a whole developed a mind, not the cells individually. In that case, the ant colony would be the person, it might not even be aware of the individual ants, and if it is, it would consider the ants just as part of its body. Unlike our body, its body would not be connected, but the colony would still consider the ant collective as a whole as its body, just like we don't consider our limbs or our eyes as separate beings. This is a hive mind: A single mind that spans the whole colony, rather than being an individual property of single ants. Note that with a pure hive mind colony, the ant would not have a mind on its own, just as our neurons have no mind on their own.
Of course there might be a continuum between the extremes, just like there's a continuum on the social behaviour of animals. One might e.g. think of a sort of hive mind that can split into individual minds and reunite. Or one might think of individual minds with telepathic contact that goes beyond mere communication, so that they may have literally shared thoughts. Or there might be both a hive mind and individual minds which interact with each other.