This paper presents DiffMoog - a differentiable modular synthesizer with a comprehensive set of modules typically found in commercial instruments. Being differentiable, it allows integration into neural networks, enabling automated sound matching, to replicate a given audio input. Notably, DiffMoog facilitates modulation capabilities (FM/AM), low-frequency oscillators (LFOs), filters, envelope shapers, and the ability for users to create custom signal chains. We introduce an open-source platform that comprises DiffMoog and an end-to-end sound matching framework. This framework utilizes a novel signal-chain loss and an encoder network that self-programs its outputs to predict DiffMoogs parameters based on the user-defined modular architecture. Moreover, we provide insights and lessons learned towards sound matching using differentiable synthesis. Combining robust sound capabilities with a holistic platform, DiffMoog stands as a premier asset for expediting research in audio synthesis and machine learning.