The key difference is that breakeven ignores losses to the surroundings, which do not contribute to heating the fuel, and thus are not able to make the reaction self-sustaining.
In comparison, man-made reactors are far less dense and much smaller, allowing the fusion products to easily escape the fuel.
A thermonuclear weapon uses a conventional fission (U-235 or Pu-239/241) "sparkplug" to generate high pressures and compress a rod of fusion fuel (usually lithium deuteride).
The fuel reaches high enough pressures and densities to ignite, releasing large amounts of energy and neutrons in the process.
[14] NIF estimates that the laser supplied 1.9 megajoules of energy, 230 kilojoules of which reached the fuel capsule.
[14] While the experiment fell short of ignition as defined by the National Academy of Sciences – a total energy gain greater than one – most people working in the field viewed the experiment as the demonstration of ignition as defined by the Lawson criterion.