Minimum spanning tree vs Shortest path What is the difference between minimum spanning tree algorithm and a shortest path algorithm?

In my data structures class we covered two minimum spanning tree algorithms (Prim's and Kruskal's) and one shortest path algorithm (Dijkstra's). 

Minimum spanning tree is a tree in a graph that spans all the vertices and total weight of a tree is minimal. Shortest path is quite obvious, it is a shortest path from one vertex to another. 

What I don't understand is since minimum spanning tree has a minimal total weight, wouldn't the paths in the tree be the shortest paths? Can anybody explain what I'm missing?

Any help is appreciated.
Consider the triangle graph with unit weights - it has three vertices $x,y,z$, and all three edges $\{x,y\},\{x,z\},\{y,z\}$ have weight $1$. The shortest path between any two vertices is the direct path, but if you put all of them together you get a triangle rather than a tree. Every collection of two edges forms a minimum spanning tree in this graph, yet if (for example) you choose $\{x,y\},\{y,z\}$, then you miss the shortest path $\{x,z\}$.

In conclusion, if you put all shortest paths together, you don't necessarily get a tree.