Halogenation of alkenyl boranes
I remember learning that these two reactions proceed via different mechanisms. I'm going to use an alkenylborane to illustrate them.

Organoboranes are electron-deficient and electrophilic; when hydroxide is added together with iodide, the first step is the formation of an anionic ate complex. This is now nucleophilic via the $\ce{B-C}$ $\sigma$ bond, and can therefore react with the electrophilic iodine:



This mechanism is somewhat analogous to the hydroboration-oxidation $\ce{H2O2}$ step, where the carbon "nucleophile" is intramolecularly transferred from boron to an electrophilic oxygen atom. This step is an intermolecular transfer of carbon from boron to iodine instead.

Incidentally, I remember that with alkylboranes the migration occurs with inversion of stereochemistry, but I cannot seem to find a reference. Given the premise of the question, I would hypothesise that alkenylboranes react with retention of stereochemistry (which would not be surprising since stereochemistry at $\mathrm{sp^2}$ and $\mathrm{sp^3}$ centres, and the stereoelectronic requirements for inversion/retention, are entirely different).



As for the other reaction, hydroxide isn't added first, so iodine (electrophilic) has to react with the double bond (nucleophilic). The subsequent steps are best described by a diagram instead.



Again I am a little frustrated at not being able to find a reference, but I am fairly sure this was how I learnt it. I literally looked up all reactions of $\ce{BR3}$ with iodine on Reaxys but couldn't find anything on it, so...