Like why use RTGs (Radioisotope thermoelectric generator) to power small spacecraft instead of just placing batteries inside them to power stuff?
Energy density for:
NiMH C battery -> 237,073 Joules per Kilogram.
Plutonium 238 (used in RTGs) -> 2,239,000,000,000 Joules per Kilogram.
Even if we assume that only 10% of a RTG weight is actually Plutonium, then we still get about 9,400,000 times as much energy available as heat from an RTG as from the same mass of batteries.
In most deep-space missions, landers, and rovers, heat generation is essential to maintain spacecraft function. However, for electrical power, the best conversion efficiency from RTG thermal to electrical power is about 7%, making it "only" 661,000 times as much energy available as electrical power from a RTG as from the same mass of batteries. That's still a pretty huge difference!
Source: Energy Density, wikipedia
RTGs are used in a very small number of spacecraft. They are used only when there are no other options, i.e. for long missions too far away from the Sun to make solar panels feasible.
Those missions have requirements for a few hundred W of power continuously over a decade or more. If you were to use chemical batteries to supply that much power, your spacecraft would become too heavy to launch.
Nuclear decay is just simply the most energy dense fuel there is. This is enough to overcome huge inefficiencies in power conversion. We can even ignore the inefficiencies of alternate storage methods, and still conclude that fissile material will store more energy per unit mass.
For RTGs I'll refer to Wikipedia's article about rtg's that have previously been or are in service. By their nature, the output and efficiency of an RTG is complicated to compute, so I'll refer to the actual measured power output, and assume the output declines along with the nuclear decay (half life of 87 years for 238Pu).
Evaluating the RTGs used in aerospace applications, the absolute worst performer was the SNAP-3B generator with a specific power of 1.3 W/kg (at launch). This was used on the Transit4B satellite, which was operational for roughly 1 year (accidentally destroyed by nuclear test detonation). During this time its 2.1kg RTG produced roughly 23.558 kWh of electricity. This gives a specific storage of 11.2 kWh/kg
Typical quoted values of Lithium ion specific power capacity are usually around 100 - 200 Wh/kg, however this post (linked article no longer accessable; see wikipedia) from the electronics stackexchange explores the performance of lithium-air batteries (currently have the highest specific energy of any chemistry) with a value of 1.7kWh/kg for a lithium air battery (Li - O2)
As you can see the absolute worst performing RTG is still several times more energy dense that the best performing chemical battery.
As far as chemical energy generation goes, fuel cells are much better, as you can more or less ignore the weight of the fuel cell and only consider the fuel (making the assumption that the mass of fuel is much larger than the cell). Hydrogen fuel cells can reach near 85 - 90% theoretical efficiency from the reaction 2H2 + O2 -> 2H2O (40 - 60% in practice). Even ignoring the efficiency loss (because it's small and I don't want to add the calculation step) we can calculate the specific energy density to be 3.73kWh/kg using the enthalpy of formation of water (the absolute theoretical amount of energy released when hydrogen and water combine)
Even fuel cells at above theoretical maximum performance are not quite as good as one of the worst RTGs (keep in mind we're only considering space applications. RTGs for land use have lots of extra radiation shielding and are very heavy)
Simply put, RTGs last a long time. Space probes need a reliable, long-lasting power source, since we can't just change the batteries when they run out. An RTG can run for decades with relatively little reduction in power output, unlike a traditional chemical battery.