What's the difference between /t̬/ and /ɾ/ in American English?

Both sounds are denti-alveolar, made by the tongue contacting somewhere between the top of the teeth and the alveolar ridge. The difference is in the manner of articulation. /t/ and /d/ are plosive, which means that the vocal tract is blocked for a moment by the tip or blade of the tongue. In contrast, /ɾ/ is a tap or a flap, with the tongue making much briefer contact and not fully blocking the vocal tract. (The respective Wikipedia entries on /t/ and /ɾ/ give more detail.)

Some dictionaries simplify /ɾ/ as /d/ to represent it's a different sound in a way casual readers will recognize. However, to use your own transcription of water, there's a definite difference in sound between

• /ˈwɑː.t̬ɚ/ or /ˈwɑː.dɚ/
• /ˈwɑː.ɾɚ/

Compared to /ɾ/, /t/ or /d/ put more stress on the middle of the word. As an American, I can say that just fine, but it sounds like I'm really emphasizing the word. Within the many dialects that do this, /ɾ/ may feel like a more neutral stress, suitable for a consonant that appears in the middle of a word as my tongue is moving from one phoneme to the next (often from a vowel to what is commonly spelled [-er]). I can do that with many words: totter, bitter, lottery, fetter, patter.

/ɾ/ tends to appear in the middle of the word. /ɾ/ with the first or last sound of a word (ten; kit) would sound odd:

• /ɾɛn/
• /kɪɾ/