We study the asymptotic behavior of key network statistics in graphon-based random graphs, including subgraph counts and the largest eigenvalue. We show that both exhibit a fundamental dichotomy in their limiting distributions. Depending on the graphon, the limiting distribution may be Gaussian or have additional non-Gaussian components. For subgraph counts, we derive the joint asymptotic distributions and develop a multiplier bootstrap method for valid inference in both regimes. Using this bootstrap together with a procedure for detecting regimes, we construct joint confidence sets for any finite collection of motif densities, providing a general framework for inference based on network moments in the graphon model. For the largest eigenvalue, we establish analogous dichotomous behavior in the limiting distribution determined by the degree function of the graphon. These results uncover a unifying structure underlying fluctuations of combinatorial and spectral network statistics in graphon models.