I found this question here: OLEDB v/s ODBC
Which gave me more information, but did not really answer the question I\'m asking, so I shall proceed from there.
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ODBC is a C API for accessing databases. There is a standard for it, it is supported by every major database vendor, it is very well documented, it is cross-platform. OLEDB is a similar interface that uses Microsoft's COM technology instead of the C API. This means that it is only easily useable on platforms that support COM.
At the end of the day, both libraries provide roughly equivalent basic functionality. Indeed, Some OLEDB drivers actually make use of ODBC rather than native database libraries.
So, if you are C# developer, working on Windows, OLEDB is the obvious choice between the two. If you are using C (or C++ not using COM), or need cross-platform support, then ODBC is the better bet.