Given the same set of nodes, why is it (generally) easier to find a Euler cycle than a Hamilton cycle?
Suppose you are trying to find a Hamiltonian cycle on a graph.  

You can solve this efficiently in the manner you describe (i.e. converting vertices into edges and then solving for an Euler path) if and only if the graph is a line graph.

There are a number of characterisations of line graphs.  

For example, a graph is a line graph if and only if it does not contain any of these 9 graphs below as a subgraph:



An alternative characterisation is that a graph is a line graph if and only if you can partition the graph into cliques such that every vertex belongs to exactly two cliques.

This characterisation is perhaps more useful as you can apply Euler's proof to these cliques.