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So I have seen that both of them can form a syllable on their own but I don't know the difference between them.

/ɹ̩/

it is a syllabic R and can form a syllable on its own as in [ˈdɔːɾɹ̩] ("daughter" in Chicago accent)

/ɚ/

it is an r-colored vowel and can form a syllable for example in [ˈdɑ̟ˑɾɚ] (Standard US pronunciation of "daughter")

I also read the answer to this question (Difference between IPA ɚ, ɹ, and ɝ) but didn't understand.

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TLDR: In terms of phonetic symbols for American English, no. These represent the exact same phonemes.

There are several ways of making an /r/ in American English, which are audibly almost indistinguishable. One of them is a "bunched r", which is made at the back of the mouth. One can use this method to add r-color to nearly any vowel. The other way is a syllabic [ɹ̩], where [ɹ̩] is a voiced alveolar or post-alveolar approximant.

My impression is that originally [ɚ] was intended to represent the bunched /r/, and [ɹ̩] the voiced alveolar approximant. But these two sounds are very close and individual Americans use one or the other or both, with really no discrimination between them, so it was decided that there wasn't any point in using two distinct symbols for two different ways of making the same sound, and [ɚ] and [ɹ̩] are now considered to represent the same phone. The website you've linked to clearly is making some distinction between [ɚ] and [ɹ̩]; possibly it is using the first symbol for a bunched r and the second for the voiced alveolar approximant (so they presumably can tell the difference between these two sounds; of course, they're trained phoneticians).

You can also add a hook after any vowel, such as [ɑ˞], [ɛ˞], [o˞], [ɔ˞], [ʊ˞] to represent an r-colored vowel. Personally, I use both ways of making /ər/, but I also use the bunching technique for the vowels /ɑr/, /ɔr/, and /ʊr/. The Wikipedia page on r-colored vowels says that you can also produce these r-colored vowels using a "retroflex articulation", but I'm not quite sure that I believe it—I certainly can't do it; I can only use the "retroflex articulation" to produce /ɚ/ and /ɝ/ (which are the same sound in American English, the difference being that /ɝ/ is stressed).

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  • I wonder why you've written '$' on either side of the 'r' in r-colored. – Decapitated Soul Feb 24 at 14:54
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    @Decapitated Soul: a mistake; it's one way to italicize on stackexchanges that use MathJax. – Peter Shor Feb 24 at 14:55
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    My impression is that the trend is that syllabic [ɹ̩] is more often used when it precedes a vowel, and [ɚ] almost universally when it occurs word-finally. Incidentally, I believe [ɚ] is very rarely used in British English transcriptions, whereas syllabic [ɹ̩] is. An example would be in the word gathering. However, there is nothing principled about any of this. As you say the two are entirely interchangeable. This is a problem with the transcription system, as can be seen from the fact that this question was asked here in the first place! – Araucaria - Not here any more. Mar 7 at 1:00
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In my opinion these two sounds are similar in that they are both only slightly rhotic, and they are dissimilar in that the second, as the symbol shows, starts as an allophone of schwa, while the first does not, or, if it does, this allophone is barely perceptible (again, as the symbol would tend to confirm), the sound being apparently the beginning of an r.

Wikipedia - rhotics In phonetics, rhotic consonants, or "R-like" sounds, are liquid consonants characteristics of rhotics Another suggestion is that rhotics are defined by their behaviour on the sonority hierarchy, namely, that a rhotic is any sound that patterns as being more sonorous than a lateral consonant but less sonorous than a vowel.

Britannica Liquid, in phonetics, a consonant sound in which the tongue produces a partial closure in the mouth, resulting in a resonant, vowel-like consonant, such as English l and r. Liquids may be either syllabic or nonsyllabic; i.e., they may sometimes, like vowels, act as the sound carrier in a syllable. The r in “father” or Czech krk “neck” and the l in “rattle” are syllabic; the r in “rim” and the l in “lock” are nonsyllabic.

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  • If the sound starts as a schwa, it should properly be represented in IPA as [əɹ] and not [ɚ]. – Peter Shor Feb 24 at 15:16
  • @PeterShor Doesn't it sound at least a little like schwa to you? – LPH Feb 24 at 15:21
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    [ɚ] and [əɹ] indeed sound very similar, and both are used (most likely by different speakers) in American English for words like serene. But [ɚ] is a single phone (the sound doesn't change in the middle) and [əɹ] is two phones ([ə] followed by [ɹ]). – Peter Shor Feb 24 at 15:53
  • They are both only slightly rhotic? I don’t understand. – Araucaria - Not here any more. Mar 6 at 11:25
  • @Araucaria-Nothereanymore. In this context, one speaks of r-coloured vowels (or rhotic vowels); by "slightly rhotic" I meant only slightly r-coloured. (R-coloured vowels) – LPH Mar 6 at 11:57

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