When children exceed the limits of their seats, the potential for injury or death is raised because their delicate heads are no longer cradled and protected in the seat shell. What happens in a crash is that the seat dumps down and back, going just as fast as the car was going prior to the crash. The belts stretch and the child's head launches forward, and when it hits the seatback in front of it at that speed, with that force, the seatback simply isn't a soft landing, it's like hitting a brick wall or being dropped from a third story window onto the ground.
So that's why you want to use a seat that fits your child... a taller convertible will keep that head cradled and protected in a plastic shell lined with foam (like a helmet) so the injury risk is much less.
Any child can die in any crash, not all crashes are survivable (no amount of plastic or foam or steel can compete with a Semi truck running a red light and hitting your side door, for example...) but hopefully that helps you see the reasoning behind why you want to keep your child in a seat appropriate for their size. (And, the infant seat may still be fine for your kiddo, if they are under the weight limit, which you can find on the sticker on the side, and has at least an inch of space over their head/are at or below the height limit...15 months means a fairly recent peds checkup, so just compare the weight and height they got to the carseat labels or manual).
Oh, and here's a graphic of injury risks... RF kids have way higher head injury risk, very likely due to the misunderstanding that kids must be RF only in infant seats and leaving kids in them too long...
http://carseatblog.com/34/kim-im-here/)