“Name -like ‘$PartialName’” is a double-quoted string; any variables inside of it get expanded. The single quotes inside that string are just text; they don’t receive any special treatment from PowerShell, and don’t affect the behavior of variable expansion.
The reverse applies as well: ‘Name -like “$PartialName”‘ is a single-quoted string. It doesn’t matter that the string happens to contain double quotes; the variable is still not going to be expanded.