Mineral ownership gives someone the right to mine on private land without paying royalties to another individual or government.
Under the Mining Law of 1872, if a mining claim on public land is patented, ownership of that land is transferred from the government to the individual (or company); this is typically on done for metallic deposits. Once patented, the owner of that patent has access to the minerals on the land as well as surface resources. Patenting of public lands for mineral extraction is a controversial subject, and is increasingly less common.
Mineral leasing gives someone the right to mine on private or public land, but they typically pay a royalty or fee to the land owner (i.e., either based on the % of the mineral produced or sold, or the units of mineral produced or sold).
On public lands, mineral leasing is common for fossil fuel mining or extraction, and for mining of some industrial minerals. For metal mines, mineral deposits are claimed (or patented) instead of leased.