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 path, to, 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 name.deb.
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.