This is a very good question and we get it all the time. I will try to be as clear as possible.
The short answer is that, in the US, there is no difference between the two objects because, most of the times, we refer to the piece of furniture one stands behind to deliver a speech or make a presentation of any sort.
The longer answer is that in fact, the lectern is the piece of furniture used for speeches and presentations. What do the experts say: The Merriam-Webster definition is: a stand used to support a book or script in a convenient position for a standing reader or speaker (here). The MacMillan Dictionary writes: a tall piece of furniture with a sloping surface where you put an open book or document when you are giving a speech (here).
A podium is usually the object with 3 different levels where athletes would stand to receive their gold, silver and bronze award. A podium is also a word used to define a stage, a large platform that is higher than the floor an audience would be sitting.
What do the experts say: The Oxford Living Dictionaries definition is: a small platform on which a person may stand to be seen by an audience, as when making a speech or conducting an orchestra (here). The Cambridge Dictionary writes: a raised area on which a person stands to speak to a large number of people, to conduct music, or to receive a prize in a sports competition (here).
It seems that "lectern" is the right word to use. So, why do we find "podium" used to identify a lectern? It comes, this is my personal point of view, from the expression "podium stand" where the "stand" is a piece of furniture, a support, used to hold pieces of paper for a speech and "Podium" the place where you are when using the stand. "Podium" was sort of an adjective for the word "Stand". It defines what kind of stand one is talking about. With time, it was contracted to only the word "Podium".
On our store, we use both words to identify the same object (a piece of furniture used to give speeches and make presentations). The Editors at Merriam-Webster themselves say that "A Podium Is the Same Thing as a Lectern".
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