Biochar is produced from sustainably sourced biomass and is used for non-oxidative applications in agriculture (e.g., in soil) and is also discussed as a raw material for industrial processes. By definition, it is used for carbon sequestration. While, the activated carbon is produced from any carbon source (fossil, waste or renewable) and engineered to be used as sorbent to remove contaminants from both gases and liquids. Therefore, the terminology depends on use of the materials.
There is an ongoing discussion on the terms and the differentiation between them. I guess the paper from Hagemann et al. (2018) gives a nice overview including a graphical decision tree for a differentiation.
Hagemann, N.; Spokas, K.; Schmidt, H. P.; Kägi, R.; Böhler, M. A.; Bucheli, T. D. Activated Carbon, Biochar and Charcoal: Linkages and Synergies across Pyrogenic Carbon’s ABCs. Water (Switzerland) 2018, 10 (2), 1–19
Activated carbon is the conventional carbon material. Biochar is belong to the group of AC. Pristine biochar has limited ability and capacity for environmental application. Researcher are working on activating biochar as engineered or designed biochar for various applications.