Twitter isn't a social network in the way Facebook and Google+ are. In fact, Twitter's user numbers fall in between Facebook and Google+ - Facebook 750M, Twitter 200+M, Google+ 18M. Twitter also doesn't consider itself a social network. It actually calls itself a 'real-time information network'. That doesn't change the fact that its users expect certain things from Twitter as a network. Here's some things that Twitter needs to do to bring itself up to social networking standards:
Create a Like / +1 with ☆ Favorite.
Probably the most underutilized feature of Twitter, the Favorite is relegated to bookmarking items right now. It can still stay that way but once you Favorite an item it should show up in the Twitter Stream and notify the tweet creator. It should also exist as metadata underneath the tweet. i.e. 3 of your friends favorited this + 5 others - it would be 3 of your friends mini-avatars and a link on 5 others that if you mouseover it you see who else favorited it.
Facebook has taken the common word 'like' and owned it. Google+ has done the same thing with the geek term '+1'. Twitter has not one, but TWO mechanisms to denote affinity - the star and the word favorite. For their purposes both can be interchanged. The only other one that is left is a heart/love, but that's moot since Twitter already has something in place.
Attribute tweets to sources in addition to original tweeter
Tumblr currently does this by pulling the source of the post and places it at the bottom. For some posts it also pulls the favicon, which makes it look more legit. Twitter could do this to any links shared. i.e. if www.nytimes.com/link/more-link is shared then it should be easy enough to recognize that the domain nytimes.com is the source and their favicon.ico should be able to be pulled and placed in front of their name. They can also pull the page's Title as another option to identify the source. This can be placed next to the timestamp under the tweet - 16 minutes ago via nytimes.com. This becomes more crucial when shortened URLs that don't expand the tweet (like t.co) hide the original source. This leads to less credibility and possible spam/phishing. By adding a source link at the bottom of all tweets with links you also free up space in the tweet; I currently will attribute a tweet to NY Times by either saying /via @nytimes or placing (@nytimes) before the link. This uses 13 characters too many in a 140 character environment.
Publish my @mentions in a new tab on my profile
If someone leaves me a message on my Facebook wall then others can see it. If someone does leave you a message on your Facebook wall and modifies the privacy settings then only certain people can see it. If Twitter were to implement this then it's version would be much simpler - if a private user leaves you an @mention then only the people accepted by that user can see their @mention of you on your Twitter wall. I often want to know what other people are saying about the people I'm following or looking at their profile. Currently you have to do a search for that info.
Show my conversations / media shared with an account
If I go to a person's profile I want to see my past conversations with them in the sidebar. The sidebar already shows the accounts I follow that follow the person and accounts that are similar to that person. What would make it more noteworthy is if I could see all of the conversations I had with them, all of the photos and videos we shared together, any of the hashtags we participated in, any of the events we shared etc. Twitter is already showing rich multimedia info from many sources and tracking all of your conversations. Why not show them? Facebook already does this when you go to a person's page.
Create a photo/video/link area
I want to see all the photos and videos someone has shared. This is standard for both Facebook and Google+. I can go to their profile and view their uploads. If Twitter can recognize when your account shares rich multimedia it can also sort it into visible areas on your profile. It already does this for your RTs and tweets. This will be even easier when Twitter's native photo uploading feature goes live for the whole network (Sidenote, why no native video feature Twitter? Your power users have smartphones where video is native.)
As far as links are concerned, this could be handled a couple of different ways. They can either be shown according to timeline in a tab on your profile or on the sidebar sorted by source. The links are the least of my concern and there really isn't a precedent for this so I'm more bearish on this than everything else.
Retweets
The retweet tab on your home page is wonderful but why can't I see it on an individual's profile? I can see how many people RT'd a tweet if I click on it. I should be able to see that metadata underneath the tweet on their profile (see Favorites for how I handled this). There should be a tab on their profile that shows their RTs. That lets me know what other people find interesting of theirs. The RT is the Facebook/Google+ Share. The RT currently shows up when your friend RTs something, but what about if a tweet is trending amongst your friends? That brings me to...
Newsfeed
Facebook currently uses an algorithm that takes into account how often you comment / like someone's post and surfaces posts that could be more relevant to you. Google+ tried to do this with posts you commented on that got more comments, but that's the wrong approach. Just because a lot of people are commenting on something you commented on doesn't mean you have an affinity for it. Now let's say a lot of the people I interact with often are chatting about something then that should surface to the top. Let's relate this to Twitter.
There are services that track who I influence and who influences me on Twitter. What this means in layman's terms: who I talk to, who I retweet, who I favorite, who I've listed, who I've private messaged (DM) and who has done all this to me. I should see more items of who influences me ranked according to what influences me the most to the least. This could be on a separate home screen tab next to Timeline, just like Facebook's Newsfeed. It could be called Affinity or something like that.
Trending
Trending topics are interesting and sometimes keep me up to date on current events, but I have no clue which ones my friends are involved in. Trending topics need to have a section that shows me what interests my friends and what conversations they're having in TT so I can join in their conversation.
Brands
Twitter currently treats brand accounts similar to regular accounts (unless they're a part of the Promoted program). Facebook makes the distinction between a personal account and a business page, as does Google+. Facebook used to list the Pages you Liked but now that feature is gone. Instead they use these insights in their ads to show you ads of pages your friends have liked etc etc. Twitter is trying to figure out how to make their Promoted Tweets relevant. One way that these could be done properly is to promote a tweet of a brand that your friend has followed along with saying which of your friends follow the brand, which have RTed the message and which have Favorited it. This would create 4 separate Promoted Tweet products - Promoted Tweet with followers, Promoted Retweet, Promoted Favorite and Promoted MegaTweet (which would include all of the above in the metadata - intrinsically the most powerful of them all).
People like relevancy. There are already "ads" in the sidebar i.e. Promoted Account and Promoted Trending Topic. This is the same for Facebook. If Twitter mimics Facebook's mechanism of how Liked content gets shared in the stream then Promoted Tweets could work really naturally. Essentially, a Promoted Tweet would be just like a Retweet from the person you follow. You didn't ask for it yet you got it because you follow them.
It seems that Twitter is starting to take small steps to bringing their network up to speed with other social networks. It recently added native photo upload ability, albeit still in a small rollout. They are also rumored to be rolling out a status update box for people's profiles, much like how you can leave a post directly on someone's Facebook wall, with this you would be able to leave an @mention on someone's Twitter wall... or would you? Even though you'll be able to send them an @mention from their profile it won't show up on their profile. I addressed this above and hopefully they'll implement a public facing version of these Twitter "wall posts" as well as some of these other points. I love Twitter and want to see it become a more full featured network.