The main point why shooters get a Picatinny scope base [in other words picatinny rail] is to achieve compatibility with all scope rings, that use picatinny base for mounting. That is the short answer: Shooters get Picatinny scope base to make the rifle compatible with scope rings, and as a consequence to install a scope to their rifle.
But that only brings us to another question - why to go for scope rings that use the picatinny base. We will go through the advantages of Picatinny scope base below, one by one.
Picatinny base is universal
The first point is that picatinny rail is universal solution.
As long as there is picatinny rail / scope base for your specific rifle, you will be able to use the same scope rings for picatinny, as on any of your other rifle.
Thanks to that you can get one top quality riflescope and set of scope rings, and use it on many rifles. As long as they have picatinny base installed, it will be fully compatible.
Picatinny rail allows easy mounting
The picatinny rail (scope base) was developed as a consequence of evolution of weaver rail. The design is made from the beggining in order to suit the needs of the most demanding shooters.
One consequence of that is that the picatinny base allows extremely easy and quick installation of any scope rings. You just need to put it on, fit it between two blocks of the rail, and fixate it. And then it just works.
Compare that to the scope rings for dovetail mounting, and you will see the difference.
Wide range of compatible scope rings
And one another important point is that there are simply many more types of scope rings that use picatinny base.
Below we go through some of the types of scope rings. Keep in mind that those are only very general categorizations. And the very same product can belong into quite a few groups at the same time.
One piece scope rings
These scope rings are the most popular ones.
They are very simple for mounting, for removing as well as for transportatoin.
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You can find these with all sorts of fixation systems. The scope rings on the picture above use screw for fixation.
Two piece scope rings
Compared to one-piece rings, these are little bit less popular, but if you need to adjust the position of the scope rings for whatever reason, these are the ones to go.
Similarly as with the one-piece scope rings, these can come with different mounting solutions. The rings above have QD mounting. You simply push the lever and you can demount the ring from the picatinny base. But you can find also fixation by screws and by another methods.
Ergonomical QD scope rings
The difference from another QD rings is that the QD levers are here in such a way, that you can demount it even in winter gloves.
So not only you do not need any tools to mount and demount them, but you can do it even with winter gloves on your hands!.
Classic QD scope rings
Classic QD rings, similarly as the Ergonomical QD rings, do not need any tools for mounting and demounting.
Compared to the ergonomical rings however, these have the lever smaller and more hidden. You will still be good just with your hands to install & uninstall.
Tactical scope rings with offset mount for red dot
The main selling point of these scope rings is that it allows you to switch between scope [for long distances] and red dot sight [for short distances].
You will find these features useful especially in tactical shooting, where you may need to switch from long distance to short distance quite often.