Sometimes these differences are discussed in terms of focus. For instance, Michael Northcott's The Environment and Christian Ethics identifies three broad approaches. Those who emphasise the intrinsic value of the non-human world and regret its destruction or transformation by humans he calls ecocentric. Those who emphasise the damage to human society represented by ecological degradation he calls humanocentric (others use the term anthropocentric, which keeps the Greek etymological theme). Those who emphasise God's glory and delight in the created order such that destructiveness is an affront to divine purposes he calls theocentric. These are not necessarily mutually exclusive and particular thinkers may draw upon multiple lines of thought. Each will lead in somewhat different directions at certain points, but the main difference lies in how they analyse the problem of ecological degradation. Why is it wrong for us to be clearing the rainforests, to be emptying the oceans of fish or to be dumping over 100 million tonnes of plastic each year? Is it because we lose species and damage ecosystems that are beautiful, unique and irreplaceable? Or because we're undermining our ability to feed and clothe ourselves, because the cost of replacing the lost ecosystem services is a drain on human society, because we're running up an ecological debt we can't possibly repay and so driving off a cliff? Or are they wrong because they represent a human attempt to uncreate, a perverse parody of God's original work?
Personally, I think any answer that doesn't draw on all three strands is likely to be deficient and lead to a poor response. Pure theocentrism could give the impression that as long as our hearts are in the right place, it doesn't matter if our actions are any benefit to our neighbour (human or non-human). Pure ecocentrism might imply that humanity itself is the problem and that any human modification of the "natural" order is wrong. Pure anthropocentrism risks becoming instrumentalist, and ignores the fact that God pronounces the created order "good" prior to the creation of humanity. These are caricatures, but all three reasons can find a place in an account that is attentive to the holy scriptures.