install a program as a user or install it as a root?
What is the difference between install a program as a user and install it as a root, all differences please and what is the best?

sudo apt-get install vlc or apt-get install vlc?

Install as root, using the sudo form of the command.

Most systems will require that you install as root.

If you look at the binary directories, /bin, /sbin, /usr/bin, and /usr/sbin all the programs there are owned by root and have 755 permissions, which is to say that the permissions are read-write-execute for owner, read-execute for group members and the rest of the world. Further, these directories are owned by root and therefore all files in there should also be owned by root.

This is the common method. It is also designed so that if you have multiple users, that they can also use the newly installed program.

Can someone get away with not doing that or altering file ownership to have it not be root? The answer is yes, but it is not advised.

On Debian and all the Debian-based systems I know of it isn't possible to run "apt-get install vlc" (or install any other program that way) as root. What makes you think that's possible?
Personally, even when I install things by dropping binary files into directories I do so as root and make them owned by root (I do this for Firefox, for example) because that means that the executable cannot be changed by the user running it. Whether that actually matters or it is actually more secure I could not tell you but to my mind executables ought (in the most part) to be owned by root and used by users other than root. There are other reasons for my doing it which I suppose come down to "consistency".

Installing programs requires root permission, as they are installed for all users to use. However, you can install some (not all) programs in your /home directory for your own personal use.