There’s no difference for the person clicking the link, but there are minor differences for the person posting it, and for a hypothetical person who might read the URL but not click.
The magic number that’s common in your two examples,
13855, is the database identifier of your post. When you visit a link of the first form, your browser is redirected to a URL of the second form, with an appropriate
#anchor to the middle of the page if the link is to an answer rather than to the question. You generate links of the first form by copying from the browser URL field.
You generate links of the second form by clicking the “Share” button under a post. The other number in the first form, here
305718, is the user ID of the person who has clicked that button – which you can verify by clicking on your profile name and looking at that URL. That user ID is used to award the Announcer, Booster, and Publicist badges. So if you’re motivated by badges and gamification, you might prefer to click the “Share” button instead of copying the URL.
A person who is reading the link, but not clicking, can divine the content of the question from the first form, because it includes a version of the title of the question. However, if the title of the question is edited, the old links will still work. The only valuable part of the long-form link is the question/answer ID; everything after that slash is ignored. That means that you can in principle deceive people about the content of the link with the long form: the URL
https://physics.meta.stackexchange.com/questions/13855/im-a-big-dumb-poopy-head also redirects to this question, even though no one here is a big dumb poopy head.
A person who is reading the link, but not clicking, can determine your user account on the site from the short form. So if you were to send an email to your boss that said
hey this SO question solves our problem
then your boss could, if motivated, visit your Stack Overflow profile page, notice from your list of linked accounts that you sometimes write long answers on Physics during so-called “working hours,” and give you grief about that. Someone who was more concerned about privacy than about Announcer badges might strip their user ID from the short-form links.