The distinction the Wikipedia article might be trying to make is between three types of algorithms:
1. Algorithms which go over all possible solutions, choosing the optimal one.
2. Algorithms which go over a subset of all possible solutions, chosen so that the optimal solution belongs to the subset.
3. Algorithms which go over a subset of all possible solution, without the guarantee that the optimal solution belongs to the subset.
The first two types of algorithms produce the optimal solution, while the third type aims to produce a "good" solution rather than an optimal solution. In my opinion, the distinction between the first two kinds is not so clear cut.
Let me start by giving simple examples for all three types of algorithms, in the context of shortest path (the example you give).
1. Try all possible paths. This is known as brute force.
2. Try all possible paths, keeping track of the minimum solution so far. Whenever the current path you are constructing is more expensive than the minimum solution so far, abandon it and choose another one (we imagine that the distance is computed on a segment-by-segment basis). This is called pruning.
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Look at the map, consider a few paths, and choose the best one among them. This is an algorithm for a human rather than a computer.
These examples are rather crude, and perhaps don't paint a very accurate picture. Pruning is crucial in many situations, for example in computer chess. If you're curious, look up the A* algorithm, which is actually used for shortest path.
Dynamic programming is a technique for speeding up significantly the brute force algorithm. It is somewhat misleading, however, to think of it this way. It is an algorithmic technique for solving optimization problems. You can implement pruning in the context of dynamic programming.
In the case of shortest path, here is one version of dynamic programming. We compute inductively the shortest path from the starting point to any other point of significance in the map using $t$ segments. Given the data for a certain $t$, we can compute the data for $t+1$ by enumerating over the last "hop" in any path from the starting point to any other point. When $t$ is large enough, we will have found the shortest path from the starting point to any other point. This is much more efficient than brute force, though not as efficient as some other dynamic programming algorithms. 242