Of your four sentences, only the first is entirely acceptable as it stands. The rest are little ambiguous; they may represent acceptable elliptical constructions; but they cannot be used to represent the same meaning as the first.
I’ll strip out the specific words and use abstract X and Y for the nouns and abstract A for the positive adjective, with A-er as its comparative grade. A-ness is the quality expressed by A.
- X is A-er than Y.
- ?X is A-er rather than Y.
- ?X is A-er to Y.
- ?X is A-er above Y.
Sentence 1 states that X has more A-ness than Y has.
Rather than means instead of or and not or in preference to, and Sentence 2 may be parsed a couple of different ways. Most simply, it claims that X, and not Y, has more A-ness than something else—call it Z. Under this parsing the sentence is an elliptical form of
2a. X, rather than Y, is A-er [than Z]. (Note that we have to move the rather than phrase; if we left it at the end it would mean It is Y, rather than Z, which X is A-er than.)
However, it is possible that X and Y are being compared to each other, as in Sentence 1. In that case, the sentence would be parsed as an elliptical form of
2b. X is A-er [than Y], rather than Y [being A-er than X].
In both cases, however, it is the construction A-er than which performs the comparison; rather than heads an adverbial phrase which modifies the core statement.
The rather than construction can also be used to compare two verbal phrases instead of two nouns, and in this case the construction may be split:
2c. I would rather do X than [do] Y. eg, I would rather eat ice cream than broccoli.
2d. I would rather VX than VY. eg, I would rather eat ice cream than be boiled in oil.
In Sentence 3, to is a preposition which plays no part in a comparison. It may, however, be a component of a gradable “phrasal adjective” (that’s a term I just made up) like close to. If that phrasal adjective exists, the sentence could be parsed in two ways, as two different ellipses:
3a. X is A-er to Y [than X is to Z]. eg New York is closer to Washington than Chicago.
3b. X is A-er to Y [than Z is to Y]. eg New York is closer to Washington than Boston.
(But the phrasal adjective better to is not current in English, so as it stands your own sentence is not acceptable.)
The to might also be used with the -ed participle of a verb which performs a comparison; to then would be either the head of a prepositional phrase modifying the verb or a prepositional component of a phrasal main verb—it’s often hard to draw the line between these. In any case, the participle would be in positive form, not comparative, since the comparative sense would be expressed lexically:
3c. X is V-d to Y. eg New York is preferred to Washington.
In Sentence 4, above works the same way; I can’t offhand think of a phrasal adjective with above, but here’s a use with a participle:
4a. X is V-d above Y. eg New York is ranked above Washington.
But a comparison employing a comparative-grade adjective almost always implies the construction with than in your Sentence 1; so Sentences 3 and 4 are unacceptable with better or with any other A-er form which does not form a phrasal adjective with the preposition.
Best Answer
Both constructions are grammatically sound. In the first example, in achieving is a prepositional phrase the object of which is the gerund, achieving. In the second example achieving is a present participle. One spelling and two parts of speech! This would have a clear antecedent in context. For the sake of argument, lets say the this is referring to "careful driving." Why not use that antecedent and make the sentence clearer and more active: Careful driving achieves better fuel economy.