Several things, actually.
In Puncture Vine (Tribulus terrestris), each flower produces five "goat heads". While they are still on the plant, these burs are in a circular arrangement, attached to the remain of the flowers by the "base of the skull" with the "horns" pointing out. In woofie's plant, the burs are sort of stacked on top of each other.
Puncture Vine burs have two main spines that are stout, unbranched, and unbarbed. They are designed for penetration of tough surfaces. You generally won't find them in animal fur, but instead, stuck in their feet. The burs on woofie's plant also have two main spines, but they are heavily branched and appear somewhat delicate. They look like hooking-type burs rather than stabbing-type burs, which would be consistent with finding them in horse manes or dog fur.
The seedheads of Puncture Vine are on fairly long (up to an inch) penduncles, while the seedheads on woofie's plants appear to be sessile or nearly so.
Puncture Vine is a very leafy plant with the burs concealed in the foliage. They wouldn't be exposed along the stem like that.
I could go on, but doubt anyone wants me to......