Can you hear the difference between 'Writer' and 'Rider'? Why?

Solution 1:

It turns out that writer and rider are not “indistinguishable” in much of the United States. The difference is that although both rider and writer have an alveolar flap in their middles, writer is [ˈɹʌɪɾɚ] with a raised and somewhat shortened diphthong, whereas rider is simply [ˈɹaɪɾɚ].

The original /t/ of write was enough to trigger so-called Canadian raising in the diphthong, since write has it but ride does not. This distinction is preserved in the longer versions ending in ‹r›.

Understand that although the phenomenon is called Canadian, it is by no means limited to that country, but extends to much of the United States as well.

We don’t think of the raising as being phonemic, but it is enough to disambiguate what would otherwise be homophones. In the referenced Wikipedia article, they also point out that this is what allows us to distinguish high school (the one after junior high) from a high school (one that is high).

Solution 2:

In his classic 1962 article "Phonology in Generative Grammar" (Word 18, 54-72), Morris Halle gave the dialect difference between the rider/writer distinguishers and those who don't distinguish as an argument in support of rule ordering as a part of human phonological systems. If rules modifying vowel quality differently before voiced versus voiceless consonants and the flapping rule which neutralizes the difference between t and d apply in different orders, we get different results. We do actually find dialects of English of both sorts, which we can distinguish in a natural way by formulating phonological systems for the dialects with exactly the same rules, but applying in a different order. This shows the linguistic reality of rule ordering in phonology.

This is a rough paraphrase of the argument, according to my recollection, since I couldn't find a copy on line. The article was reprinted in the 1964 collection of essays The Structure of Language.

In response to the question, no, in my own native dialect, I don't distinguish the pronunciations of "rider" and "writer". Do I hear a difference in others' speech? Sure, if there is one, since I'm a linguist.