Twitter works a bit differently compared to other social media networks relating to the links posted.
The link you share at Twitter sometimes is not validated by Twitter. That means the Twitter Cards for that tweet is not properly defined.
You can check whether the URL you are using is validated by Twitter or not on its official Twitter Card Validator
So, if the link generates a preview here, that means; it will, when shared on your Twitter profile, but if the validator doesn't crawl the meta-data and shows an error or something, then the link is not validated by Twitter itself and your content & link will be shared as a plain text tweet only.
This is nothing which SocialPilot can handle, so if the link generates a preview at SocialPilot and not at Twitter, don't freak out! Just check using the Twitter Card Validator.