homebrew (e.g. Macports) is a package manager. It allows you to manage packages (update, delete, etc.). Most importantly, homegrown will put together an application on your platform. This is especially important for ports, for example. from Linux.
homebrew will give you greater and finer control over what you set, where, which compilation attributes you want to use, etc. But this is due to more complex work and the need to know your way around the team line.
Downloading a binary file and placing it in the Applications folder is much simpler and usually works fine. If you are not a developer and do not need to manage many different tools, I would recommend sticking with binary downloads. However, if you are a developer, most likely you will not come across a package manager if you need tools such as Tex, ghostscript, ImageMagick, etc.
To answer your last question: no, the application will not appear in the Applications folder automatically (at least in the case of Macports). You will need to create a symbolic link if you want, for example,
$ ln -s /opt/local/bin/python /Applications/myPython
Edit:
@Ned Deily
Your example above does not make much sense. / Opt / local / bin / python is not an application in the sense of OS X (rather, this is what you run from the shell command line), so why would you create a symbolic link to it in / Applications ?
It's right. This was the first thing that appeared in my mind as a (possibly bad) example.
In addition, MacPorts installs application packages in / Applications using the MacPorts subfolder
Also true. But the responsibility of the package developers is to tell MacPorts about it (I think ...), so don't count on it. I went and installed Inkscape through MacPorts. This installs runnable, which needs the -g option to run with a GUI (but I did not have X-Server ...). So no luck installing the application package in applications.
@ user2426316
The short answer is yes, it is possible to have both a compiled binary and an application package. To be more precise, I would install one or the other. Depending on how the application was written, the two installations may interact unexpectedly with each other. It is possible that both files are shared configuration files and, therefore, even “know” about the changes made to another instance. But this behavior can lead to overwriting if the developer has not been careful. On the other hand, they can use a separate set of configuration files, and if you need some time to configure the application, you will have to do it a second time in another instance.
Another thing to think about is that both instances are likely to receive updates from time to time. Will you always sync instances? Is this possible (maybe there is no new application package that includes the same changes)?
In general, I see more trouble and, frankly, no good (except for eye candies, maybe). Rather, create a script to run a binary or symbolic link and assign a fancy icon to this file.