My understanding is that the Prasangikas who follow Gorampa completely deny the validity of conventional truth. For them, conventional truths are completely false objects appearing to an ignorant mind, so when you become enlightened, you do not perceive conventional truths. Tsongkhapa teaches that conventional and ultimate truths are mutually supportive and equally valid (albeit that conventional truths incorrectly appear as inherently existent to non-Buddhas) and that Buddhas realize the union of conventional and ultimate truth simultaneously.
The difference is that for Tsongkhapa, conventional truths are able to withstand ultimate analysis since all that is being analyzed is the subtle object of negation, inherent existence, not the existence, of a conventional truth.
For Gorampa, they can not, since no phenonena can survive examination via the course object of negation, existence. Gorampa accepts that Candrakiriti specifically identifies (in the Prasannapāda) a subtle object of negation, but according to Gorampa, it is just a formal identification since inherent existence is automatically eliminated when existence itself is analyzed.
According to Tsongkhapa, what is being misperceived by sentient beings in conventional truths is the inherent existence of conventional truths i.e. he claims that when an ordinary person sees a chair, they are seeing an inherently existent chair. However, Tsongkhapa also claims that ordinary sentient beings are incapable of distinguishing between mere existence and inherent existence.
Gorampa points out that Tsongkhapa's first assertion is untrue, since inherent existences does not appear, and Tsongkhapa's second assertion is self-contradictory, sentient being only see existences, not inherent existences.
Tsongkhapa replies that conventional truths are linguistic entities, mental imputations, and that therefore, the notion of inherent existence is embedded in all imputations of conventional truth. Gorampa counters that this interpretation of conventional truth is faulty, since in fact relative truths are first and foremost appearances to a deluded mind, and what such a deluded mind grasps is not a truly existent object, but rather a merely existing object, and imputations of inherency are confined to the philosophical speculations of scholars, not the naive imputations of ordinary persons such as Chai wallas, who would never imagine their tea cups had some intrinsic nature that made them teacups.
So, at base, a large part of the disagreement hinges on how these two masters understand conceptual operations in sentient beings and what they understand Nagarjuna, Buddhapalita and Candrakirti to be saying about such conceptual operations. This is why Tsongkhapa places such importance on seperating and identifying the correct object of negation, and why Gorampa thinks that such an effort misses the point and is unnecessary, since the coarse object of negation is sufficient for removing wrong views via the classic tetralemma (in ordinary persons -- awakened persons have no need of the caturskoti).