Why does FeCl3 react with carboxylic acid, but not with sulphonic acid in this question?
It all made sense because $\ce{CH_3COOH}$ was an acid

$\ce{CH_3COOH}$ being an acid facilitates the complex formation as the chlorido ligands can be easily removed as $\ce{HCl}$. But, that's not the only reason why the red colored complex $\ce{[Fe(OAc)_3]}$ was formed.
You have to understand that the stability of the complexes is also determined by how well the ligand interacts with the central metal ion.Here, the ligand forming the coordination complex is $\ce{OAc^-}$ and this is associated with enough negative charge density to stabilize the complex formed.

So, I felt that the compound being acidic due to the sulfonic group would work similarly as the first compound and thus react with $\ce{FeCl_3}$. However on verifying with the answer key, I found that it would not react with $\ce{FeCl_3}$

This is because although as you have said has a sulfonic acid group, it is not the acidic strength of this ( or acidic strength of $\ce{CH_3COOH}$ in the first question) that drives the complex formation.It mainly depends upon the strength of the ligand and effective nuclear charge of the central metal ion.The ligand in this case, is associated with lot of electron withdrawing groups like $\ce{-NO_2}$ and that makes this a poor ligand.Also, the steric hindrance due to the bulky groups also has a role to play for the complex formation not to take place.
NOTE:
All my arguments made regarding stability of complexes is as per the Crystal Field Theory which assumes the interaction between metal ion and ligand to be purely ionic.That's why, I was comparing the charge density of the ligands in the 2 cases.