The two variants look for the package in different locations (unless the current directory is the root directory).
sudo apt install ./path/to/package/name.deb
looks for a package file named
name.deb, in the subdirectories
package, starting from the current directory (
.). This only works if the current directory contains a directory named
path, which itself contains a directory named
to, which itself contains a directory named
package, which contains the file
sudo apt install /path/to/package/name.deb
looks for a package in the directory
/path/to/package, i.e. the same sequence as above, but starting from the root directory, not the current directory.
Put another way,
./ isn’t part of the
apt syntax, it’s part of the file’s path, and interpreted as usual for paths.