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Is there a difference between the two when used as in the following sentences?

The extent of the disaster was initially underestimated.

vs.

The magnitude of the disaster was initially underestimated.

Both seem very similar, but I haven't found either as a synonym for the other in a Thesaurus (only as related words).

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They're at not synonyms. Magnitude is about the size, extent is about the spread in the context. Think 'severe but localized, versus 'mild but widespread.' –  Kris May 26 at 13:25
    
@Kris completely agree. extent:magnitude :: further:farther –  njboot May 26 at 14:39

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The Longman Dictionary of Contemprary English says, for extent,

how large, important, or serious something is, especially something such as a problem or injury; It is too early to assess the full extent of the damage.

and for magnitude,

the great size or importance of something; They did not seem to appreciate the magnitude of the problem. (and also) the brightness of a star; the force of an earthquake

So both magniutde and extent can apply to size and/or importance.

The difference is somewhere else: it lies in the adjective 'great' (Latin 'magnus'); with extent, the people estimated the size and/or importance of the disaster to be great or small – we do not know – but it was greater than they tought; with magnitude, the people estimated the size and/or importance to be great and it was even greater than they thought.

Earthquakes and stars, even if their magnitude is small, still involve incredibly large/great amounts of energy…

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