Both the water cycle and carbon cycle are Biogeochemical cycles, that is, the material moves between the biotic (living organisms) and abiotic (lithosphere, atmosphere and hydrosphere) compartments of the planet Earth.
Both of these are nutrient cycles, by which the nutrients (water and carbon) move across the various compartments of Earth.
Both the cycles use plants as an integral component. The water cycle uses plants for uptake of water and transpiration from the surface, so that water can be sent to the atmosphere. Carbon cycle uses plants for uptake of CO2, by which it is removed from the atmosphere (and sent back to atmosphere when the plant is dead). Water is retained in body of plants and is used for various purposes, including transport of nutrients and in metabolism. Similarly, carbon is used in photosynthesis and also form the body of the plant (through various organic molecules).
Both the water cycle and carbon cycle use subsurface for long term storage. The water, when it enters the sub-surface, forms groundwater and can be stored for long duration. Similarly, carbon is stored in rocks (preferably as carbonates) for long term( and this is actually being thought of as a viable alternative for carbon sequestration for preventing climate change).
Hope this helps.