I promise this is an actual, answerable question. But I want to explain myself when I call these specific words "weird"; English is so often "exceptional" that referring to any particular part of it as being weird seems unnecessary. These comparative words have bothered me for a while.
Than is a bit of an odd word in English. Wiktionary (I don't have access to the OED) gives two definitions; one is a conjunction and the other a preposition. I want to focus on its use as a conjunction.
First of all, than is bizarrely "active" when compared to other frequently-used conjunctions. I'm afraid that I don't know the technical term for this, but I think it makes sense: and just connects two or more things, but disconnects (well, negatively connects I suppose) two or more things, or and nor express alternation, and yet is similar to but although I'd say yet is a bit more disconnecting. (And actually has an "active" form, seen in statements like "try that and you'll get shot!", wherein and is used to express some sort of causation. Or has a similar negative usage: "don't run quickly or you'll fall". But note that negating the second part forces and: "don't run quickly and you won't fall".) If is similar in this respect: it too has a function of "introducing" a kind of statement. In a language that doesn't have a lot of morphology, like English, these words are often necessary to form certain kinds of statements, whereas in more marked languages they may not be. In Latin, the "ablative of comparison" is a construction wherein the comparative form of an adjective is used to compare two nouns, one being in the nominative or accusative case, and one being in the ablative case, as in
Catō est Cicerōne ēloquentior
'Cato is more eloquent than Cicero'
(very literally, 'Cato is, than Cicero, more eloquent'); in this usage the use of the ablative for expressing "movement away from" is extended to stating that two things are different as though they were moving apart (source). Of course, Latin quam is also used for comparisons, having a very similar function to English than; we'll see more about quam later.
So than has a lot to say. I place it in the same class as (some uses of) as, so, and even like (though than is a bit of an outlier). These all can be used to compare two or more things; the difference is that than (although one use of as fits too; see below) can contrast (what I'll call disjunctive comparison) while as, so, and like are used more to connect two or more things. But the fact that they're used to compare is not the only reason that these conjunctions/prepositions/adverbs are unique. They seem to be special in that they accept much more ellipsis than other conjunctions. Take for instance (1):
(1) She does that much better than I do.
We'll speak of a "left side" and a "right side" of such statements as (1): in this case, the right side "I do" is technically an abbreviation of the predicate of the left side "She does that". (There's a third part of the sentence, which we'll call the "comparative part"; in this case, it's "much better".) Therefore the right side, when "expanded", is I do that. In practice, the right side is actually frequently more abbreviated than it already is:
(2) She does that better than I/me.
The choice of the nominative (I) or oblique (me) forms to use in statements like (2) is traditionally informed by what exactly the right side is saying. Here, since the right side is contrasting something with the subject of the left side, it is regarded as more correct to have the right side be in the same nominative form. But then there's (3), where the right side is not really abbreviating anything:
(3) He ran faster than ten miles per hour.
The right side of (3) can hardly be understood as ellipsis for "ten miles per hour ran". This is the prepositional use of than. Accordingly, pronouns should be in their oblique form when used as such; but here we have a problem. If we're comparing two pronouns with than, it could often go either way:
(4) A ran faster than B.
If we're comparing the speed as an attribute of A's running with that of B, then A, as the subject of the sentence, should be in the nominative form, and B, the object of the preposition than "should" be in the oblique form as in (4.1).
(4.1) I ran faster than him.
However, we could also be comparing the manner of A's running with that of B; in this case (4.2) is "more correct".
(4.2) I ran faster than he [did/did run].
This is far too fine of a distinction to make most of the time; I can't think of any examples where it would actually matter. The "erroneous" use of the conjunction than with oblique pronouns seems to have stemmed from reinterpretation of than, being commonly placed directly between the two things one is comparing and also different from the other conjunctions, as always being a preposition and thus putting the nouns in their oblique forms (as is done with prepositions).
Now we need to turn to as, so, and like. By all accounts these are usually prepositions, although often extended into conjunctions. This is true of several words, and indeed many conjunctions in many languages started out as prepositions. As has many meanings. Wiktionary gives as a definition as an adverb, conjunction, and preposition. They all have a similar underlying meaning of "while" or "at the same time"; I believe that it is from this that the comparative meaning is extended. In (5) we have an modification of (4) with the as...as coordinate pair instead of than:
(5) A ran as fast as B.
Even though the meaning has changed from a disjunctive comparison to one that equates the left and right sides, the problem of determining the function of B is the same as in (4). B might stand for the subject of run, and "should" therefore be in the nominative; it could also be the object of a preposition. In this case, the issue is further compounded by the fact that as appears twice. It seems that the first and second as are really different words -- or at least different senses of the same word. As is also a synonym of so, and so has a comparative sense on its own; (6) is the title of a movie, but it illustrates this well:
(6) As above, so below.
In (6), as and so also coordinate: "[just] as [it is] above, [it is] so below". Obviously this is quite a poetic turn of phrase, but it is valid; in older writing the as...so construction is quite frequently encountered, in forms like that of (7.1):
(7.1) As X is Y, so also Z.
(7.1)'s right side ("also Z") shows ellipsis of is in the left side; the "full" form would be as in (7.2).
(7.1) As X is Y, so also is Z.
So can also stand on its own, while still retaining the meaning; however, it becomes part of an independent clause as a result (because it no longer has support from as):
(7.3) X is Y; so also Z.
The function of as in (7) at first seems to be expressing cause, and replacing it with because should produce a statement that is as valid, but it doesn't really.
(7.4) *Because X is Y, so also Z
I wouldn't be surprised to find this, necessarily, but it sounds odd to me. This emphasizes the way in which this group of words is different from the other conjunctions: because tolerates ellipsis to a much lesser degree than they (haha). Because of this, the whole issue of case doesn't apply.
Like is similar, but less versatile. Its ellipsis-accepting comparative usage seems to be less, and more recent (Wiktionary has an interesting (though unsourced) bit of information about this) than that of the others. It can often replace as when it isn't paired with another as:
(8.1) He runs as I do.
(8.2) He runs like I do.
There is a slight difference in meaning though: (8.1) seems to be more "absolute", comparing his running's existence to that of my running, while (8.2) compares his manner of running with mine; however, in context, they could both have the same meaning:
(8.3) When we race, he runs as I do.
(8.4) When we race, he runs like I do.
Note that as, so, and like, while ellipsis-tolerant, are less so than than. (9.1) is correct, but (9.2), (9.3), and (9.4) are less valid ((9.2) sounds the most correct to my ears).
(9.1) He runs faster than I.
(9.2) *He runs as fast as I.
(9.3) *He runs like I.
(9.4) *As he runs, so I.
In other languages, the behavior of comparative words is similar. Latin quam was mentioned above, but Latin's ways of comparing two things are extensive. Quam corresponds closely to English than. Latin, being heavily inflected, requires the left and right side to be in the same case, regardless of which out of the five/six/seven they be. This is similar to English, although English only marks pronouns for case, so it's hard to determine whether a word is a subject or an object out of context. Latin also has tam, which, when coupled with quam, provides an as...so or as...as-like meaning.
Spanish has que 'than' and tan...como 'as...as/as...so' for comparisons: eres más bello que yo 'you are more beautiful than I' and no conduzco tan rápido como tú 'I do not drive as fast as thou dost' (just to keep things unambiguous). From examples I could find, it seems that you will most often find the right side in the nominative form. Further examples (from Wiktionary) illustrate this (translations slightly altered): no es tan alta como nosotras 'she is not as tall as we', ¿tienes tanta hambre como yo? 'art thou as hungry as I?'. It would be incorrect to say *no es tan alta como nos or **¿tienes tanta hambre como me?* (and I'm sure would sound pretty weird to a native speaker). Spanish also adheres to the "copulae requires nominative case" rule: ¿quién soy yo? 'who am I' and not **¿quién soy me?* '*who am me'. Latin, of course, does this too.
I don't know enough about other languages, but I'd imagine the other Romance languages behave similarly.
I promised that this is a real question. Specifically, I would like to know for what reason these words (the group as a whole, a subset, or just an individual one) behave like this. What makes them so special? How did they come to mean what they do today (what is the history of these words)? I've written all I can say about their current use; I'm very interested in their history but I do not know of any sources for this ("conjunction development", I think, is sadly under-studied).
(Than and so have a similar origin as a form of an ancient pronoun; since this is also true of Latin's quam and tam I suspect that asking anything about this semantic development would be off-topic and more for a general linguistics site. As apparently derives from a contraction of also, which makes it an outlier here.)