The difference is simple yet important. An around strategy side steps the problem and so you don't have to deal with it. In other words, the problem is still there but it no longer impacts your goal. Through tactics on the other hand deal with eliminating the problem by moving through it. Think of it as the difference between taking a detour and drilling a tunnel.
Neither of these tactics is better then the other but each does work better for certain types of problems. An around solution is great for one time problems, those things that block us from getting a specific project accomplished and for large problems that we do not have the resources to tunnel through at the moment.
On the other hand the recurring problems in our lives should be dealt with head on using a through tactic. This can take more energy and even derail a project for a time while we build the tunnel. However, in the long run, if a problem is one that keeps reoccurring in our life, the time spent going through will save us time down the line when we do not have to deal with the problem any more.