While there is much history to be learned about the subtle nuances, let's agree on a few key broad strokes that will prevent you from making a fashion faux pas.
Blazers were born from sport jackets. Blazers have been around since the days of English cricket clubs thus, they had an emblem or crest denoting to which school or club you belonged. Blazers are usually a solid color and made of smooth fabric. Depending on your blazer style of choice and what you pair it with, blazers are easier to dress down than sport jackets.
Sport jackets have been "in" since the mid-nineteenth century. Sport jackets are textured and oftentimes patterned. They are generally a looser fit for ease of movement and, per tradition, have three or more buttons.
In the end or modern times, the lines have blurred together because the pictures here are of two different Zegna jackets that look just about the same to me- one is advertised as a blazer and the other a sport jacket. For style sake, if you're in a pinch grab the jacket of your nicest suit along with a haute patterned shirt, and you'll be fine just about anywhere!.