Image credit goes to tumblr.com
First let us understand some different terminologies, and some basic science behind all topics!
Height & Bone Growth 
The fastest period of growth is before birth, with the baby growing from almost zero to a length of about 50cm in nine months. This speed falls after birth with an average growth of 5.5cm per year at eight years old. During puberty, growth speeds up again. This is called the pubertal growth spurt. Before puberty, boys and girls grow at similar speeds, but during puberty boys grow more than girls. The average height of an adult man is 14cm taller than the average height of an adult woman.
Bones increase in length because of growth plates in the bones called epiphyses. As puberty progresses, the growth plates mature, and at the end of puberty they fuse and stop growing. The whole of the skeleton does not stop growing at the same time; hands and feet stop first, then arms and legs, with the last area of growth being the spine. Growth slows down and stops when a child has gone all the way through puberty and has reached an adult stage of development. This means that growth does not stop at a particular age, but children who are ‘early developers’ will stop growing before late developers. After the growth plates fuse, there is no more increase in height, and we all then shrink gradually as we get older.
Normal growth is controlled by a number of hormones:
- Growth hormone, made in the pituitary gland, which is the most important factor
- Thyroid hormone
- Sex hormones, testosterone or oestrogen, which are important for growth during puberty
Nail Growth 
At 20 weeks in the womb, humans suddenly sprout tough little casings from the tips of our tiny digits. By the time we’re born, our fingers and toes are crowned by fully-formed nails that will be with us for the rest of our lives. Over the ensuing decades, the average person will devote hundreds of hours to carefully clipping, painting and filing these structures. But beyond this dedication to aesthetics, few of us ponder the purpose of our nails.
Why do we have them, and why do they grow?
Most of us do know that nails are made of a tough, dead substance called keratin, the same material that makes up hair. But nails actually start out as living cells. Behind the cuticles on fingers and toes, just beneath the skin, a structure called the “root” churns out living cells that go on to form the nail. Also known as the matrix, this little pocket of flesh connects to blood vessels, which supply the nail with the nutrients it needs to make new cells. [Do Hair and Nails Keep Growing After a Person Dies?]
How do nails grow?
Nails grow out of deep folds in the skin of the fingers and toes. As epidermal cells below the nail root move up to the surface of the skin, they increase in number. Those closest to the nail root get flat and pressed tightly together. Each cell becomes a thin plate; these plates pile into layers to form the nail.
As with hair, nails form by keratinization. When the nail cells accumulate, the nail pushes forward.
The skin below the nail is the matrix. The larger part of the nail, the nail plate, looks pink because of the network of tiny blood vessels in the underlying dermis. The whitish crescent-shaped area at the base of the nail is the lunula (LOON-yuh-luh).
Fingernails grow faster than toenails. Like hair, nails grow faster in summer than in winter. A nail that’s torn off will regrow if the matrix isn’t severely injured.
Difference Between Nails and Bones 
Nails are simply dead squamous cells. As these cells keep replacing themselves Nails keep ‘growing’.
Bones on the other hand are made of dense connective tissue.
They stop growing once the epiphyses fuse (Epiphyse is the growing end of the bone.)