In soccer, you will often hear these two terms- finishing and shooting- interchangeably. But they have very distinct meanings that all players must understand. Having the ability to shoot, does not mean you can finish.
I'm not trying to minimize the skill involved in finishing versus shooting. Both attempts on goal require players to utilize particular mechanics and techniques that players must practice regularly and independently. Through practice, players learn where and how to strike the ball to create the type of shot they want to produce. another difference includes how hard players strike the ball- shooting will often require power, whereas finishing often requires finesse.
For my teams, shooting also serves a secondary purpose. Shooting from distance early in a match can force defenses to step forward and defend higher up the pitch. Therefore, I will often coach my teams to take shots from 20+ yards away from goal in order to force the defenders to worry about shots from distance. This creates space behind the defenders for attacking players to run into and create chances for finishing on goal.
Finishing requires players to make decisions quickly in tight spaces and requires various techniques and surfaces of the foot or head- inside/outside/laces. Due to the limited time often afforded players in front of the ball, they must practice one-touch and two-touch shooting in this space. These are often scrappy goals that develop from mistakes and scrum-like conditions.
The final key, in both shooting and finishing, is going to be repetition. The higher the number of reps a player takes, the more they will perfect their technique and mechanics. It isn't as easy to do during indoor, but I would often advise my players to either stay after training sessions or spend time independently working on shooting and finishing- this may include taking free kicks and corner kicks.
Types of Shots