If you Smite it you get mana back. If you deal the killing blow to it you get Crest of the Ancient Golem from it.

Using Smite:
Using Smite upon the Blue Sentinel will restore a large amount of mana over 2 seconds (25% of your maximum mana). This bonus can only be claimed once per spawn.

Dealing the Killing Blow
This unit regens 25 flat mana regeneration per 5 seconds and 0.5% of their maximum mana (or energy) per second and has 10% cooldown reduction on their abilities. If slain, this buff transfers to the killer.

Gaining mana by smiting the Blue Sentinel is something they added in S5. It is handy for jungle sustain for mana heavy users, though most junglers tend to reserve smite for other jungle camps unless they are in dire need of mana to complete their jungle route or deal with an immediate emergency elsewhere on the map.

Most junglers will take blue buff on their first clear, for the gold an XP and cool down reduction. Mana users will of course enjoy the mana regeneration aspect of the buff.

Mana intensive junglers may continue to take the buff throughout the game if they need the cool down reduction or the mana regeneration.

Donating Blue

However, if you have a mana using mid the usual practice is to donate the second and later Crests to the mid laner. There are some non-mana using mids such as Zed that greatly benefit from the cooldown reduction so donating the buff to them can be beneficial, though this practice is more common in team environments than soloqueue.

Some mid-laners stack CDR so there may come a point in the game where they are no longer benefiting from the 10% CDR that the Crest provides. You may still choose to donate it to them in the case of heavy mana users. Ziggs would be one such example.

As an alternative to donating the buff to the mid laner or keeping it for yourself, you may choose to give the buff to another champion on your team for either the CDR or the mana regeneration, though typically you would shoot for offering it to the champion that would benefit from it most. E.g. someone like Ezreal who has fairly high mana costs and who usually doesn't itemize much if any CDR.

If during the laning phase the person you are donating the blue buff to is getting killed constantly and simply donating the buff over to the enemy team you can choose to stop giving the buff to them. This can be a tricky situation morale-wise though. More game aware mid-laners will actually tell you to keep the buff if they are losing lane badly when you ping them to come over and take it.

Once laning phase is over (generally considered to be when the outer lane towers go down) and team fights start you should again start to donate the blue buff to the mid laner despite their score, unless they prove to be utterly incompetent (people have bad lanes) This is because the more spell rotations they can get off in a team fight the more damage they are going to do or the more CC they will be able to apply.

Damage Over Time Effects

Something to watch out for when getting leashes for jungle camps in general or donating buffs is damage over time effects of which their are several in the game. Teemo and Blinding Dart, Twitch and Deadly Venom, the Bladed Armor defense mastery.

Some junglers choose to forgo Bladed Armor simply because of the risk of having the DoT effect from it doing the last tick of damage to the buff creature "stealing" it from the champion you are trying to donate it too.

You can ask champions like Twitch and Teemo to only autoattack the target twice rather than the three times that are typical for a proper leash because their poison damage will roughly add up to a third autoattack.

Other:

Blue Buff is important enough to the flow of the game that you should keep tabs on your blue buff and the enemies' blue buff so you can be there to take either when they spawn. Arranging buff steals is a bit outside the scope of what I'm trying to accomplish at the moment with this article, but I may create a post about it once I have a better understanding around the parameters for contesting for the enemies' blue.