Update 23 October 2016: There’s a pretty easy to implement workaround using IFTTT. Full instructions here: How to display Instagram pictures correctly in tweets using IFTTT.
Update 26 April 2016: Thanks to @thameswatch who pointed me to a December 2012 (!) article explaining that Instagram pics haven’t been shown in-tweet since then, and explains the reasons behind it.
If you share a picture on Twitter from Instagram… it doesn’t show up as a picture on Twitter. I’ve no idea why that should be.
You need to click the link to go to the Instagram website to see the picture there and not as an autopreview on Twitter. I find this reasonably odd as I thought Twitter and Instagram were best buds but perhaps not. I’m fairly sure Flickr images autopreview, as do animated gifs, Vines etc (remember you can switch off autoplay) and YouTube videos.
You can see this in action by looking at the @Instagram feed which includes links to Insta images, none of which show up (other images on their feed do show up but these were uploaded directly to Twitter and are pic.twitter or other autopreviewing services).
All the instagram.com/ links are pictures but they don’t show up as such (I’ve tried on desktop browser Twitter, Twitter for iPhone, Echofon for iPhone and Janetter).
By contrast here’s a Vine, showing up perfectly well
Here’s a tweet I sent with a picture that I uploaded to Twitter via my phone, it shows up fine (because I’d not used Instagram, though I did add it to my Insta account).
Anyone know why? Seems a bit of a mystery.