The left one is just a common way of attaching buttons to uniforms - often military. It meant that the buttons could be easily and quickly replaced as necessary. The back lug (loop shank) goes through an eyelet-type hole and is held is place by the split pin. Australian military uniforms certainly used this type of button until well after WW2. They were also widely used on other types of uniforms - fireman's, transport workers, hunt buttons on big UK estates - but they were also widely used for dress buttons - that is, fancy buttons that were used on men's dress shirts. Also fine buttons for women - often in precious metals or with gem stones. A nice way of showing off.
I would think the one on the right is made of lead. My mother had several similar buttons in her button box. They could have been homemade, in a mould - buttons for poor people. Before the mass-produced plastic buttons of the 20th century, well-off people used buttons for decoration and display, poorer people used them for utilitarian reasons, and made them from whatever materials were to hand - fabric, wood, shell, and sometimes molten metal if they had the means. They were considered valuable and not to be discarded - hence the button box or jar. Lead buttons were said to have a possible secondary use as sinkers when fishing.
Buttons are a lovely collectable - there are many books written about them and lots of interesting Internet sites.
