What we commonly call wingtip vortex is technically not a vortex in the strict sense. The spiraling circulating air around the wing tip that trails behind the wing is actually the product of two processes:
1. Trailing Vortex Sheet
From the OP, you may know that a lifting surface sheds trailing vortices, everywhere along the trailing edge. The ensemble of these infinitesimal trailing vortices forms a vortex sheet behind the wing trailing edge. You may also know that a vortex line induces a circulating flow field around it. When you sum up all the effects of the trailing vortices, you can compute the flow field around and behind the wing at every point.
The graph below (Ref Drela, Flight Vehicle Aerodynamics) illustrates the trailing flow field around the wingtip. As you can see, there are two cores of circulating flow centered around each wingtip, exactly as we would see from flow visualization.
From the drag perspective, the above analysis is completely identical to the lifting line analysis for a high aspect ratio wing: the induced drag is the only drag you're going to get out of it.
2. Vortex Sheet Roll-Up
In the preceding discussion, we've been assuming that the trailing vortices trail inline with the far field velocity; i.e. we have a rectangular trailing vortex sheet. But that's not completely accurate. From flow visualization, we can see the wingtip "vortices" trailing up and away from plane.
This can be understood by the roll-up of the vortex sheet itself, as it interacts with not only the flow field, but also neighbouring vortices. This is illustrated below (Ref. Drela, Flight Vehicle Aerodynamics).
For a high aspect ratio wing, the effect of the roll-up is negligible and we need hardly concern ourselves with it when estimating lift and drag. The roll-up becomes a lot more important for low aspect ratio wings, such as a delta wing, as illustrated below (Ref. Anderson, Fundamentals of Aerodynamics):
In the case of a delta wing, the vortex roll-up induces strong suction behind the leading edge centered around the vortex core, which results in much increased drag, as well as lift, than if the vortex roll-up is absent.