The graph on the left has a Hamilton path (many different ones, actually), as shown here:

$The Petersen graph with a highlighted path visiting every vertex exactly once. The path starts at the top outer vertex, proceeds down and two the left to the bottom-left outer vertex, then to the top-left outer vertex, then in and across to the top-right outer vertex, and finally down to the bottom-left outer vertex and up to the bottom-left inner vertex.$

The graph on the right does not have a Hamilton path. You would need to visit each of the “outside” vertices, but as soon as you visit one, you get stuck. Note that this graph does not have an Euler path, although there are graphs with Euler paths but no Hamilton paths.