I believe that they're almost the same thing, but hashHistory adds a few extra characters to a route while browserHistory has a clean root. For example, if you had a route to an about page like this:
<Route path="/about" component={AboutComponent} />
And you were using hashHistory, then the route in your web browser may look like localhost:8080/#1231/about (or something similar), while with browserHistory, the route in your web browser would look like localhost:8080/about.
However, when you're using browserHistory, refreshing the page on one of the routes causes an error, so you have to modify your start script. It should look something like this:
"start": "webpack-dev-server --inline --content-base . --history-api-fallback"