So, what makes a cue an American pool cue? Well for a start the cues used in the game of American pool are a different style and weight to English cues. The game of American pool is played on larger tables and with larger and heavier balls. Correspondingly the cues have a bigger tip and are slightly heavier to reflect this. Typically the tips of American pool cues are between 11 and 13 millimetres in diameter to give a better "grip" on the larger cue ball, and are around 20-22 ounces in weight instead of the lighter English pool cues which are somewhere around 17 - 19 ounces in weight.
Another subtle difference is in the style of play in the American game. American pool players often like to wrap their finger around the cue when playing a shot so this has led to cue shafts that are parallel or the same diameter for the first couple of feet to give a more comfortable grip on the cue whereas the English style of playing gives rise to a slightly tapered shaft all the way down the cue.
And then of course being American the decoration on the cues is a lot more flamboyant. They tend to have what they call linen wraps around the butt end of the cue which are basically linen thread tightly would around the cue butt and glued in place which gives a very good grip to the cue. These are brightly colored and make for stunning looking cues and decorations on the grip. The downside is that on the cheaper cues the wrap can unravel, making the cue look very tatty.
An American pool cue made by the great cue makers like Balabushka are collectible items these days fetching many thousands of dollars, so check who made your cue before you toss it aside and get a new one.