What is the difference between an anti-inflammatory and an antibacterial property?
Anti-inflammatory properties settle inflamed areas, while antibacterials kill off bacteria. Inflamed areas are responses to damage, while bacteria cause damage.
Anti-inflammatory drugs are those that settle an inflammation, which is an area of red, raised,rash-like, often sore tissue. An inflammation is actually the body trying to heal an area by inflaming the tissue around it to protect it, so you should be quite certain that a particular inflammation is doing more harm than good before you take anti-inflammatory medicine.
Antibacterial drugs are those that specifically target bacterial pathogens. Antibiotics are antibacterial and work by breaking down the cell membrane and effectively bursting the bacteria. With no real structure or protection, the bacterium dies very quickly.
Inflammation is not the same as infection. Infection might cause an inflammation, but they are not synonymous to any extent. Antibacterial drugs may reduce inflammation by destroying the pathogen and therefore the need for inflammation, though infections are not necessarily accompanied by inflammation.