When should I use Kruskal as opposed to Prim (and vice versa)?

Use Prim's algorithm when you have a graph with lots of edges.

For a graph with V vertices E edges, Kruskal's algorithm runs in O(E log V) time and Prim's algorithm can run in O(E + V log V) amortized time, if you use a Fibonacci Heap.

Prim's algorithm is significantly faster in the limit when you've got a really dense graph with many more edges than vertices. Kruskal performs better in typical situations (sparse graphs) because it uses simpler data structures.

I found a very nice thread on the net that explains the difference in a very straightforward way : http://www.thestudentroom.co.uk/showthread.php?t=232168.

Kruskal's algorithm will grow a solution from the cheapest edge by adding the next cheapest edge, provided that it doesn't create a cycle.

Prim's algorithm will grow a solution from a random vertex by adding the next cheapest vertex, the vertex that is not currently in the solution but connected to it by the cheapest edge.

Here attached is an interesting sheet on that topic.

If you implement both Kruskal and Prim, in their optimal form : with a union find and a finbonacci heap respectively, then you will note how Kruskal is easy to implement compared to Prim.

Prim is harder with a fibonacci heap mainly because you have to maintain a book-keeping table to record the bi-directional link between graph nodes and heap nodes. With a Union Find, it's the opposite, the structure is simple and can even produce directly the mst at almost no additional cost.