Being a non-user of social media, of which I do not consider SDMB to be a part, I am not clear on a particular style of reporting that I’m seeing quite frequently. In an article on some subject, the author will quote, word for word, what, say, Trump said in a tweet. And then, below this, it will print the same thing as a sort of copy of the tweet. Why both? What’s the point? Does it have anything to do with the fact that I read actual newspapers and many articles are meant for smart phones? What’s behind this strange new style?
It could be that some of their publishing avenues (such as email newsletters or browser-based news readers) don’t include images. Services for the or sight-impaired also would not be able to get the text out of the images, as they are just text readers.
Showing an image of the tweet is proof that it exists/existed so it’s beneficial to show it (although they can be doctored so…)
To avoid impropriety they can embed tweets in their articles, not as an image but actual live text from Twitter itself. But tweets can be deleted from Twitter and if you have an embedded tweet that gets deleted (or even if the embedding fails, and it’s not shown in your article) having the text below it preserves the content of the tweet for the purposes of your article.
Also it might be the editor’s policy that images are not quotes, quotes can only be text. An image adds proof but doesn’t “count” as an actual quote.
The actual image is useful to the audience - I wouldn’t be surprised if they see it as more “real” and they can know there wasn’t any text that was left out to make it more sensational. The text is searchable, which is great for people like me who want to look up things sometimes several months later.
Whatever the reason, and I suspect it’s a combination of fallback for images not showing and a way of confirming the quote’s accuracy, it’s bloody annoying.
For me, at least, when such pages are loading, the plaintext usually loads a lot quicker than the Tweet-box.
We’re all posting on a text-only message board, so if the text weren’t written out it would impossible to copy it here for a quote.
Often, I cannot read the small print of the screenshot. I’m glad to have it typed out
Text-to-speech systems may not work on images. Nor does the Lynx browser, which is text-only.
What used to be a straightforward situation - i.e. if you post a picture, include an explanation, both for people whose vision is not clear and for people whose browser isn’t showing pictures - has become a confusing mess, due to a number of factors. I’m not even sure what those factors all are, but whatever they are, they sure have proliferated. 
Are the Twitterboxes actually images? It seems to me that they’re usually text, too, just formatted differently.
IME the twitterboxes are embedded tweets, served up by Twitter itself. So it contains text, images, hyperlinks, etc. just like the original tweet.
That used to be more common than it is now. A few news orgs who embedded tweets in their stories found out the hard way that the tweeter can change the tweet - not the content of the tweet, or the twitter handle, but the display name. So The Sun and the Daily Mail in the UK found such messages as “DON’T BUY THE SUN” and “PAUL DACRE IS A CUNT” right in the middle of articles on their websites. So they tend to use screenshots now.
Yes, this is tied up with it. Other notes in this thread point to the social eagerness, to the publishers of any old blog to big ones, even to producers of television shows, who when this whole Tweet business came out they could show they were techno-hip and in touch with A Man in the Street. And, as mentioned above, one can “prove” (supposedly) someone (the Tweeter) said something most easily, at one level, by printing it.
As to Zipper’s comment here.
There’s journalism, business, and law. Every single thing that someone writes in a Tweet, prints in it, belongs to Twitter. Now, when President Trump tweets, he is communicating from/as a non-private citizen or corporate identity. It’s unlikely the White House would or could (I’m not up on Pentagon Papers-type law, or anonymous source cases).
You’d think every time a Web outlet embeds a Tweet it would be a winner for Twitter (free exposure/instant ad) but they lose money having to serve and refresh that embed every time someone reads it in the embedding page. The hope is that the reader of the embedded page will click into the Tweet and sign up.
ETA, on this point:
If The NYTimes quotes a Tweet from Not A Famous Person of Public Interest, without getting his permission, that person doesn’t like it, he can sue the Times.
I’m sometimes annoyed by BBC’s web site when they don’t quote tweets, instead relying on some kind of imbedding that is blocked at my workplace. I don’t think it’s a simple image file because no other images get blocked (as far as I know).
To repeat: it is not “a file” (to be seen by users) at all. It is a tiny code block given by Twitter to the site, which directs that screen-space to Twitter’s computers.
If someone deletes their Twitter account, the embedded tweet would not appear and the article would not make sense. Also if Twitter is down temporarily, the tweet would not appear.
To expand on this a little, almost every ad you see on-line is a code to some-one else’s computer (ie not the host server of the website/blog) and not even to those maintained by the advertiser, but to a profit-making third party host which itself negotiates via millisecond live auctions for ad content, supplied and bid for by yet other servers.
I guess your question has been answered but I think you included a few redundant words between the word “Twitter” and the question mark.