There is a bit of confusion about which of these metals is safer for sensitive ears, or wherever. Niobium is generally available only in a chemically pure form, whereas titanium is available both pure and in a bewildering list of alloys (such as many surgical implant grades).
I list a few of the more popular grades and designations of titanium here. Most of them are hypoallergenic. Most of the jewelry that I sell is pure grade #1 or #2 titanium. Some of my ball posts are an alloy, medical implant certified, and with no detectable trace of dreaded nickel.
The etching and anodizing process strip the few percent of non-titanium elements from the surface, and then create a protective shield of titanium dioxide.
Chemically speaking, titanium is less likely to be absorbed by biological systems than niobium. Neither should cause an immune response or allergic reaction.
But when it comes to piercings, part of the issue is abrasion. Titanium is a hard metal that could have a slightly rough surface on the microscopic scale. The act of putting it in might abrade a tight hole.
Niobium is a softer metal, so the surface will yield more to pressure. This might make it better for extremely sensitive skin. The same softness is why the initially brighter colors of niobium don’t last as long as titanium colors.
Whereas I am distinctly a titanium partisan, my colleague at Wear-Earrings-Again.com prefers niobium, from her personal experience.
When in doubt, try both. If both work, then choose only by price, color and design.