Installing applications is a way to embed them in the OS. It's a kind of standard, you offer procedures like installing and uninstalling that should have the same functionalities for all applications (even "change" under Windows). Countless times I've "installed" applications with a single shell script that came with them, and then had troubles removing such programs, having to look for single files. If the programmer uses the standard of the OS to make an executable that can be installed, that won't happen. You can also easily view a list of the installed programs at any time. Under Linux, additionally, if we're talking about a package manager, it is convenient for the user to have an easy way to download and install a program by just typing its name.
Last but not least, some applications are required to be installed and recognized by the OS (for example services in Windows).