All are variations of parallelism, with two distinct wrinkles:
- Whether the determiner or article must be repeated or not
- Whether the head noun is singular or plural
All four constructions are valid, but produce slight differences in emphasis.
The difference between the old and new scheme lies in...
The difference between the old and the new scheme lies in...
In the first example, the is a determiner modifying "old and new scheme." Readers will generally understand that it breaks down into "the old scheme and the new scheme." The second example repeats the determiner and further emphasizes it. Generally, readers will still understand "the old scheme and the new scheme." In this case, the difference is slight because no other modifiers are interrupted when the determiner repeats. Fiction editor Beth Hill, in a post on this subject at The Editor's Blog, shows how repeating the determiner can mark a further difference between two items:
The flat footballs and soccer balls had been stored in the basement for a long time. (Both footballs and soccer balls are flat.)
The flat footballs and the stained soccer balls had been stored in the basement for a long time. (Only the footballs are flat—the soccer balls are stained but not flat, at least not that we know of.)
The flat footballs and the soccer balls had been stored in the basement for a long time. (We have no clues about the condition of the soccer balls. From this wording—with just the addition of the article the—we can’t assume that they’re flat.)
Repeating the is a clue that other premodifiers like "flat" may only pertain to the first noun. In your case, that is not relevant since the only premodifiers for "scheme" are already distinguished by "and."
The difference between the old and new scheme lies in...
The difference between the old and new schemes lies in...
Both work, but with some caveats.
Plural signals that there are two or more schemes, at least one is old, and at least one is new. It does not limit the potential number - there could be two old and three new schemes. If the existence of only two schemes is evident from context (you've only been discussing two schemes so far in your text), it should provide no confusion.
Singular is acceptable because readers will understand that the two adjectives are mutually exclusive, such that it is not one scheme that is both old and new, but two: the old scheme and the new scheme.
Take note: if the adjectives could both describe the same noun, this might be confusing:
The difference between the hot and wet car lies in ...
Some readers might expect "the hot and wet car" to be followed by a second item, like "the cold and dry car," since technically a single car could be both hot and wet. In this case, adding the article avoids the issue ("the hot and the wet car"), since the duplicated determiner could only pertain to a second car. (As described above, duplicating the determiner helps mark further differences between items.) Confusion can also be avoided when using mutually exclusive adjectives (e.g. user-centric and network-centric).
