Perhaps Tawney's real view is that while equality of wealth is very important, it is not the most important thing.
I take Tawney to mean that we can put goods into at least two classes. In one class are those where, if one person is to have more, than at least one other must have less.
In the other category of goods are those where at least some can have more without anyone ending up with less. At this stage Tawney gives no examples, but consider the good of "a feeling of security". If a neighbourhood feels safe, then someone moving in to that neighbourhood may benefit from an increased sense of security without anyone else suffering a cost of any sort. There is only gain.
This dual concern is a constant theme in egalitarian thought, certainly up to Bernard Williams's classic paper The Idea of Equality (1962). Williams argues for the egalitarian view of material distribution, "each according to their need", on the grounds that other policies are insufficiently governed by reason and are thus irrational.
This argument was countered with great force by the libertarian philosopher Robert Nozick. Nozick argued that egalitarian theories of distributive justice proceed as if goods fell to earth like manna from heaven; as if all consumable products existed in a "big social pot" and we should sit there waiting for our share to be allocated to us.
Consider Aesop's fable of the grasshopper and the ants. One fine day in winter, some ants were busy drying their store of corn, which had got rather damp during a long spell of rain. Presently, up came a grasshopper and begged them to spare her a few grains. "For," she said, "I'm simply starving."
The ants stopped work for a moment, though this was against their principles. "May we ask," said they, "what you were doing with yourself all last summer? Why didn't you collect a store of food for the winter?" "The fact is," replied the grasshopper, "I was so busy singing that I hadn't the time." "If you spent the summer singing," replied the ants, "you can't do better than spend the winter dancing." And they chuckled and went on with their work.
We may not all admire the ants' rather sadistic lack of charity here, but it seems very hard to say that the grasshopper has any claim in justice for a share in the ants' product, still less an equal share.
It seems to me that in the last 20 years or so, many theorists and politicians have become obsessed with the sort of issues that come to the fore in thinking about this fable.
Although he may not wish to put it exactly in these terms himself, Ronald Dworkin's position is that the requirement of egalitarian justice is to even up fortune and misfortune that are the result of good and bad luck, but not to even up fortune and misfortune that are the result of good and bad choices.
Or, to put this proposition in a slightly different way, it is a requirement of egalitarian justice that undeserved (and only undeserved) disadvantage should be rectified.
This sorts out the ants and the grasshopper nicely. The grasshopper is starving because of her bad choice of singing rather than working over the summer. Had she failed to work because she didn't have the ability to, or had the ability, but no access to land, the case would be very different. But the ants would have no obligation to, as Dworkin puts it, "subsidise her choices".
What we should appreciate, though, is how such a view would be applied in practice. You will be entitled to welfare benefits only if you can show that you lack the opportunities that others have had; this, after all, is what makes the difference in the cases where we think the grasshopper is, and is not, entitled to support from the ants.
One thing that is remarkable about this is that such highly conditional systems of benefit were once considered right-wing policies. My own view is that contemporary egalitarians have forgotten half of Tawney's teachings. Conditional benefits can create social division and humiliation. A society of equals would have less inquisitive welfare policies.