what is the difference between the usage of these two symbols? aren't they basically the same thing?
Questions about IPA "usage" ought to be off topic for the same reason that questions about usage in English are off topic. The linguistic question is whether IPA defines these two things differently (usage may then conform to that definition or deviate, depending on the author's choice).
The first thing to do is obtain a copy of the Handbook (Handbook of the International Phonetic Association: A guide to the use of the International Phonetic Alphabet), which tells you what the IPA is. The primary embodiment of the IPA is the chart, an online version being here. You can see there that the symbol [ə] is defined (called "schwa", described as the "mid central vowel", and the breve diacritic ("Extra-short) is as well, therefore [ə̆] refers to an extra-short mid central vowel. Small raised schwa is not in the chart of sanctioned symbols, although small raised h = ʰ is in the chart, under "diacritics". The symbol ᵊ is non-standard. Nevertheless, on p. 53 in the illustration of Arabic by Thelwall & Sa'adeddin, the authors give the phonetic transcription [ʕad.dᵊ] which is explained as "an incipient syllable onset with [ə]". On page 117, Okada uses unofficial combinations such as raised small s after t to denote a "unitary affricate", and various raised letters on ɹ to denote a particular variant of Japanese /r/, calling it "an affricate with short friction" (note that "affricate" is not an IPA concept). That article also has raised small d, again not in the list.
So, it is not "in IPA, but it is a practice used by phoneticians. Generally, the raised latter indicates a release-type modification of the preceding sound. A "short schwa" is a different phonological thing from a "schwa-like release of a consonant", even if they are phonetically the same thing.