Lebesgue vs Riemann

The two integrals answer the same question — what is the area under a graph — by different bookkeeping. Riemann chops the x-axis into thin vertical strips and is happy when the function does not wiggle too much within each strip. Lebesgue chops the y-axis and gathers all the x's where the function sits at a given height, regardless of how scattered they are. The second method copes with far rougher functions.

The precise comparison is clean. On a bounded interval, every Riemann integrable function is Lebesgue integrable and the two integrals agree. Lebesgue's criterion pinpoints exactly which bounded functions are Riemann integrable: those continuous almost everywhere (the set of discontinuities has measure zero). So Lebesgue is a strict extension on bounded intervals: more functions, same answers where both apply.

There is one honest place where Riemann reaches further: conditionally convergent improper integrals. The improper Riemann integral of (sin x)/x over [0, infinity) exists and equals pi/2, but the function is not Lebesgue integrable there, since the integral of its absolute value diverges. The right summary: Lebesgue dominates for absolute integration and limit theorems; improper Riemann (or its modern cousins) is still needed for conditional convergence.