How to find those files that have different names but exactly the same content?

You could install the good fdupes or you could just reinvent the wheel with bash, md5sum and awk:

find path/ -type f | xargs md5sum | awk '{
	sub("[^/]*/","",$2);
	if (cache[$1])
		print "Found: "cache[$1],$2;
	else
		cache[$1]=$2
}'

path is where you want to search for duplicates. You can limit the search with the find maxdepth option.