Both could be used. I think "that suffers the most..." has a ring of informal and less academic. Hence, found more times, way more than "that most suffers".
Examples of sentences with them, and with similar structure (albeit with other words):
Why You Should Be The One Who Loves The Most.
The Girl That Smiles the Most, Suffers The Most.
The sector that most suffers from unemployment is young people.
There are many cases of reduction of the useful life of equipments in several Industrial sectors due to corrosion, and the petrol one is one of those that most suffers from this problem.
Universities in many cultures and times have had "golden ages"; the most recent throughout the Western World lasted from the end of World War II to about 1970. Now the prevailing mood is one of pessimism: the golden age is over. In fact, however, universal access postsecondary education is healthy and continuing to grow in most nations; mass access higher education is generally static; and it is the elite sector of higher education that most suffers a decline in prestige, in faculty morale, in rate of growth of enrollments and financing, in independence, and in other ways.