If you use a screenshot it is a good idea to also put the text below it or in the picture's alt attribute, for screenreaders.

For Twitter specifically, a link to the tweet is much more genuine than a screenshot, as screenshots can be faked, where tweets cannot be edited

One small note about using a screenshot versus actual text is that a screenshot does not scale with resolution.

I think a lightly styled html box with the tweet as text would be the best of both worlds.