We may start with descriptions, where both turn out to be substances that are known to help us combat bacteria which may try to harm us. The difference between the natural antibiotics and the synthetic antibiotics is, however, in the fact that the former are products of nature (directly harnessed from the fields, being typically plant parts), whereas the latter are products of laboratory-based chemical synthesis. To get a synthetic antibiotic, you need to know which chemical combinations have an antibiotic effect (that is, a bacteria-killing effect), obtain the ingredient chemicals and mix them in the right proportions to end up with the antibiotic. To get a natural antibiotic, on the other hand, you need to know which plants (and which specific parts of them) have an antibiotic effect, and then go out into the fields to obtain those said plant parts, use them in the right way , and benefit from the said antibiotic effect.
The difference between natural antibiotics and synthetic supplements is not just in terms of definitions, of course.
Natural antibiotics differ from synthetic supplements in terms of (typical) working speeds. We tend to see the synthetic antibiotics working faster than the natural supplements. But note has to be taken of the fact that we are talking of typical cases here: for there are natural antibiotics that are known to work faster than some synthetic supplements. All the same, the typically faster working speed associated with synthetic antibiotics is the main reason as to why the synthetic supplements are commonly used in medical emergencies: where a person is already afflicted of an illness emanating from the bacterial infection, and where speedy decimation of the bacteria is essential. What is notable here is that the efficiency of synthetic antibiotics tends to be their undoing too: as most of them end up inadvertently killing the useful symbiotic bacteria as well.
Natural supplements also differ from synthetic antibiotics in terms of safety profiles. The natural supplements are seen as being, on the whole and in many respects, safer than the synthetic antibiotics. Noteworthy here is the fact that use of supplements is not always on a transient basis (although that is ideally how it is supposed to be). Rather, there are people who find themselves with conditions that force them to use them on a long term or very frequent basis. Such people, if they were to use the synthetic antibiotics, would almost certainly end up with very nasty side effects of such long-term antibiotic use. But where the milder natural supplements are used, the long term outlook may be much better.
Natural antibiotics differ from synthetic supplements in terms of (typical) working mechanisms. We are looking at a situation where the typical synthetic antibiotic works by directly cutting down (killing) the harmful as well as, inevitably, some useful bacteria. This is against a situation where the typical natural antibiotic works by not simply killing the bacteria, but also enhancing the body's natural capacity to fight off such bacterial infections in the future.