When choosing the type of barrel you want for your AR-15, whether you are looking to upgrade what you already have, or if you are building your own, it is important to take notice of what the difference actually are of the options that are available to you.
So when choosing your barrel, the most important part when choosing either way, whether it be carbine length or mid-length, it is overall length of the barrel.
The operative measurement that most manufacturers use is the measured distance from the gas port to the crown of the muzzle.
This number is the amount of barrel that remains after the bullet has passed the gas port and is then forcing high-pressure, high-temp gasses into the action. This is commonly referred to, by most operators, as the dwell time of your weapon.
So, logically, having the gas port closer to the crown or your AR-15, as it would be with a mid-length barrel system, it reduces the dwell time and consequently, lessens the total volume of gasses forced back into the action while cycling.
The advantages with a mid-length system are: a moderately reduced recoil, overall smoother operation or your weapon and reduced wear on parts. Though in so doing, you will sacrifice the tradeoff of having a very slightly reduced reliability under extreme or exceptional conditions while firing your weapon. Whether these conditions be extended use, or extremely hot or dry temperatures.
As for carbine-length systems, the dwell time, which is very much dependent on overall barrel length, can be significantly higher.
The carbine-length system will force more gasses back into the action of your AR-15 over a longer period of time, due to the barrel length much greater. This relays more force to the key and carrier, thus resulting is the cycling of the action faster, with more force and also with an higher level of reliability. Though the trade off for carbine-length systems, comes in the form of an increased recoil pulse and slightly more impact on parts wear over time.
As with everything in the world of firearms, everything always comes down to function of your specific AR-15, along with personal preference.
If you are a competition shooter, you may want a longer or shorter barrel to get the proper barrel twist rate whether you are shooting up close or long distances.
Even if you are just a basic operator, interested in learning what options are available to you to upgrade your AR-15, doing the research on each of these options is a good place to start.
The next step is understand what you plan to do with your weapon, so you can cater and customize your weapon to your specific needs. But make sure to understand where and how each different manufactures measures these systems, they are not always uniform across the board from different manufacturers.
As you can see, neither system is inherently better or worse than the other. They are simply two variations of an operating system that gives the operator one more option to tune their weapon to a specific usage. Hopefully this quick explanation has given you an idea of what those are, and will help you decide on which system is best suited for your specific weapon application.