As I know, carbon capture is usually used as a term describing a technology used to capture carbon.
Carbon sequestration and storage are often used to describe carbon sequestered or accumulated by vegetation or sediment. But there is a little difference. When it concerns about the capacity of carbon sequestration, carbon sequestration rate is used. When it concerns about the amount of carbon stored in sediment, carbon store is used.
Please refer to a paper below regarding carbon sequestration and carbon storage.
Adame, M. F., Kauffman, J. B., Medina, I., Gamboa, J. N., Torres, O., Caamal, J. P., Reza, M., Herrera-Silveira, J. A. (2013). Carbon stocks of tropical coastal wetlands within the karstic landscape of the Mexican Caribbean. PLoS ONE, 8, e56569.
Alongi, D. M. (2009). The energetics of mangrove forests, Springer, New York.
CCS describes the complete process of carbon capture and carbon storage or sequestration. Sequestration is more general. It can mean simple storage of CO2, for example in exhausted reserves of gas, injection to enhance oil recovery, or dissolving it the in the ocean. But more recently it includes also the chemical fixation of CO2 to form carbonates.
The answer from Rüdiger Grote is very accurate. Some addition on why the two terms co-exist.
Therefore, just as a (partly historical) addition: CCS (resulting in geological storage) originally referred to Carbon Capture and Sequestration, in order to emphasize the long-term or near-permanent character.
In the EU, CCS was re-coined as Carbon Capture and (geological) Storage. This was mainly done by the PR people from the Zero Emission Platform, who liked the sound of Storage better than that of Sequestration. This change of terminology was adopted by the European Commission, more or less during the drafting of the CCS Directive, if I recall well.
In contradiction to its 'temporary' character, in the EU sequestration will commonly be used for all non-geological storage (biomass, soils, etc.).
Outside of the EU, sequestration remains to be used for geological storage (e.g. CSLF). I'm so used to seeing both terms, that I can't tell if this divide is still as sharp as it used to.
I am eagre to know, how this business of carbon capture and storage is going to help and safeguard our under ground natural resources , in particular the ground water, geological stregth and qualitative aspects of flow nets within under ground aquifers ...
If you make this a starting question, I'll be happy to provide input. As background, at least in Europe there are several projects looking into co-use of the underground, including CGS and groundwater, considering both socio-economic, environmental and sustainability aspects.
Of course underground storage is not the best solution, in termes of sustainability. Better solutions are being investigated converting CO2 in useful products, such as for example in commercial plastics:
"CO2 Recovery" and "CO2 Capture" is about the same. "CO2 Storage" comes after CO2 is captured. Storage means normally storage in underground reservoirs, and also to enhance oil recovery from oil wells.
CO2 can also be fixed after capturing. For instance injecting it in greenhouses to accelerate growth of plants. Or converting it into chemicals. See more: http://alchemy.cchem.berkeley.edu/static/pdf/innovativechemistry.pdf
See also my recent article where converting CO2 by artificial photosynthesis is performedArticle AGENDA 2030: A Challenge for Chemical Science and industry