Lets back up here. I'm assuming you are using a "Pages" Index Type on both sites. So if that's wrong, please correct me.
Secondly, take a look at the documentation specifically where it talks about Analyzer Types. You stated one is "White space" and the other is "Starts with". If you read what each type is defined as you'll see the white space type will look at each individual character (includeing white spaces). The starts with type will look the group of characters searched for and match the group vs. individual characters. This difference alone will yield much different results with each site.
Thirdly, you need to review the "Allowed" and "Not Allowed" content on the site. This is where you define what is included and excluded from the index. It could be as simple as saying the whole site (/%) is included BUT only Menu Item page types. OR it could give a specific path you want to EXCLUDE with a given page type.
Event though they are different versions, the functionality and features for Smart Search are nearly identical across those versions.