Understanding climate dynamics requires going beyond correlations in observational data to uncover the underlying causal process. Latent drivers such as atmospheric processes play a central role in temporal dynamics, while direct causal influences also exist among geographically proximate observed variables. Traditional Causal Representation Learning (CRL) typically focuses on latent factors but overlooks such observable-to-observable causal relations, which limits its applicability to climate analysis. In this paper, we introduce a unified framework that jointly uncovers (i) causal relations among observed variables and (ii) latent driving forces together with their interactions. We establish conditions under which both the hidden dynamic process and the causal structure among observed variables are simultaneously identifiable from time-series data, and our guarantees continue to hold in the nonparametric setting through contextual information that recovers latent variables and causal relations. Building on these insights, we propose CaDRe (Causal Discovery and Representation learning), a time-series generative model with structural constraints that integrates CRL and causal discovery. Experiments on synthetic datasets validate our theoretical results. On real-world climate datasets, CaDRe delivers competitive forecasting accuracy and recovers visualized causal graphs aligned with domain expertise, thereby offering interpretable insights into climate systems. Code is available at https://github.com/MinghaoFu/CaDRe.