I read the definitions of both on wikipedia but they didn't mention any relationship between the two terms although there must be one because they are dealing with the same thing: spatial data. Can we consider that regional GISs are building blocks of a national SDI?
The way I see it is Spatial Data Infrasture is the whole network and framework of geospatial data. It involves everything from standards to data to databases to software. A desktop or web GIS can be a part of a SDI but it is not the entire SDI. All parts are required for the use and analysis of spatial data, as expressed in the following quotations:
The term spatial data infrastructure was coined in 1993 by the U.S. National Research Council to denote a framework of technologies, policies, and institutional arrangements that together facilitate the creation, exchange, and use of geospatial data and related information resources across an information-sharing community. Such a framework can be implemented narrowly to enable the sharing of geospatial information within an organization or more broadly for use at a national, regional, or global level. In all cases, an SDI will provide an institutionally sanctioned, automated means for posting, discovering, evaluating, and exchanging geospatial information by participating information producers and users.
The document which this is quoted from also gives a good indication on the different of a GIS and a SDI. See: http://www.gdmc.nl/publications/2001/Spatial_data_infrasructure.pdf
Steiniger and Hunter (2012) distinguish the following five components:
(i) Spatial Data (or spatial information), (ii) Technologies, i.e. hardware and software, (iii) Laws and Policies, (iv) People, i.e.: data providers, service providers, users, and (v) Standards for data acquisition, representation and transfer.
and Kuhn (2005) gives this deffintion:
"An SDI is a coordinated series of agreements on technology standards, institutional arrangements, and policies that enable the discovery and use of geospatial information by users and for purposes other than those it was created for."
Kuhn W (2005) Introduction to Spatial Data Infrastructures. Presentation held on March 14 2005. Available from: http://www.docstoc.com/docs/2697206/Introduction-to-Spatial--Data- Infrastructures
Steiniger, S. and Hunter, A.J.S. , 2012. Free and Open Source GIS Software for Building a Spatial Data Infrastructure. In: E. Bocher and M. Neteler (eds), Geospatial Free and Open Source Software in the 21st Century: Proceedings of the first Open Source Geospatial Research Symposium. Heidelberg: Springer.
ESRI 2010 - GIS Best Practices: Spatial Data Infrastructure. http://www.gdmc.nl/publications/2001/Spatial_data_infrasructure.pdf
In addition to the answer above, sometimes financial resources are also counted among the components of an SDI.
See for instance Loenen, B. van; (2006) Developing geographic information infrastructures. Avaialble from: http://www.tandfonline.com/doi/abs/10.1080/13658810701851412