Is there a difference between a Menahel and a Rosh Yeshivah, or are they synonymous?
Depends on the institution. In some, there is one rabbi who is both setting spiritual leadership (rosh yeshiva), and responsible for the mundane running of the institution (menahel), possibly including fundraising. The Netziv served both roles in the Volozhin yeshiva. Think of menahel as a school principal, or company's chief operating officer. Rosh yeshiva is more like a CEO.
In many yeshivas today, the roles are split. (Rabbi Rakeffet has noted, in retrospect, that Volozhin should have split the roles in the next generation, with R' Chaim Brisker as rosh yeshiva and R' Chaim Berlin as menahel.)
מנהל means "leader". When used with the definite article, the noun is likely referring to the highest director of the organisation in question. In the case of a yeshiva, that would usually be the Rosh Yeshiva, but doesn't have to be.
From the root נהל, see the BDB entry here.
In the colloquial ways they are used they aren't synonymous at all. Menahal basically just means a school principal. Even an elementary school has a menehal. But an elementary school without higher grades would never have a Rosh Yeshiva. In elementary school he frequently also owns the school but rarely teaches a class.
On the mesivta or Beis Medrash level he more frequently teaches a class as well. On the Beis Medrash level the menehal is always just an employee because if he owned the school he would be called Rosh Yeshiva. Even if someone else is teaching the highest level and more respected and also referred to as Rosh Yeshiva. (In such a case there could also be a subtle distinction between being called Rosh Yeshiva or Rosh HaYeshiva)
In a Mesivta without a Beis Medrash where the owner of the school has some level of direct non administrative involvement with the school he would be either be called Menehal or Rosh Yeshiva. Usually he would only be called Rosh Yeshiva if someone else is active as principal/menehal.Then he would e called Rosh Yeshiva to distinguish himself from the menehal. Otherwise a mesivta would not have a Rosh Yeshiva.
Beis Medrash always has a Rosh Yeshiva. While it possible that the Rosh Yeshiva on a Beis medrash level is also just an employee his main position is giving the highest level shiur and setting the tone in the Yeshiva. He isn't the principal. If he does not own the yeshiva where he is Rosh Yeshiva then he would not be making principal like decisions such as hiring staff, school schedule or discipline .