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).