What is the difference between ODBC and OleDB?

Question

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.

I am working in C#. I'll spare you the long story about how I arrived at this conundrum, but basically I'm trying to decide between ODBC and oleDB.

We work with a lot of different clients who have vastly varied Databases (some SQL, some oracle, some something else that I've never heard of and didn't bother to remember the name of)

Now, from what I understand, ODBC is old, and was/is the standard. And now OleDB has come along and... is... different? but accomplishes the same thing (it talks to databases)

Why would I want to use one over the other? ODBC is (according to the above post) cross-platform, which is good, but he offers very little information as to what OleDB offers that ODBC does not.

In my other research, I've found (on the MSDN forums) People saying "use OleDB if you can, if you have to, resort to ODBC" Naturally, three's no reasoning given for this, so I'd like to hear some.