I am not a big expert, but the natural compound usually have a stronger effect than synthesysed ones. But the pross is the cost of isolation / purification of the natural compound...

Antimicrobial phytochemicals may not cause resistance, but this need to be validated by in vitro and in vivo studies. However, natural may not be better. Higher dose of antimicrobial phytochemical may be needed to achieve the same effect of sythetic antibiotics.

synthetic medicines or antibiotics have more chances to developed lots of side effects, toxicity, accumulation as well as drug addictions whereas the phyto-chemical or nuraceutical are naturally available resources provided that they have very less chances of side effects, accumulation and addictions too. moreover we believed prevention is better than cure hence use of nutraceutical for prevention can be more beneficial over synthetic antibiotics used to cure.

Generally, natural products is biosynthesized and assembled inside biological system, so they have been designed to bind with biological targets, which make them better candidates for developing new drugs, regarding the antimicrobial effect, the ability of natural molecules to bind to different targets could help to overcome the bacterial resistance, also you would find several reports that address the synergistic effect of combination of natural products with common antimicrobial against several microorganism

If you consider to the mechanism of actions in both synthetic and natural antimicrobials, based on the group of synthetic antimicrobials, target of action is not specified on a special cell and lead to toxicity and alot of side effects on host cells, while, in this regard, natural antimicrobials are safe than the synthetics.

Chemical drugs usually rely on a single active ingredient, whereas medicinal plants have complex phytochemical compositions that work in synergy in vivo (italic). This complex phytochemical composition makes it harder for bacteria to develop resistance. For example, although berberine is a well-known natural antibiotic compound, it is harder for bacteria to develop resistance to Berberis vulgaris (italic) or Hydrastis canadensis (italic) taken as a whole plant extract compared to isolated berberine compound. There was also a recent study by Daddy et al published in Phytomedicine on a group of patients who didnt respond to a chemical drug artemisin but were fully recovered by taking a whole plant of Artemisia annua (italic).