Why is a BMS needed?

An LFP cell will permanently fail if the voltage of the cell falls too low, typically below 2.4V, or is pushed too high, typically above 3.8V or charged below freezing (32F / 0C).  The difference between Lead-acid batteries and LFP is that Lead-acid will eventually be damaged when over-discharged or over-charged, but only if these conditions persist for an extended period of time.  A lithium battery can be severely damaged by over-charging and over-discharging just one single time.

Surprisingly, a lead-acid battery will recover a majority of its capacity from over-discharge after it has been left in a discharged state for multiple days,  depending on battery type and brand. The BMS ensures the cells never reach these points, by cutting off the power flow into and out of the batteries.

Due to real-world manufacturing differences, the cells in any battery are not 100% identical. If a battery pack is not properly balanced, some cells will reach a fully charged state, or discharged state, earlier than others. The differences will increase if the cells are not balanced/equalized regularly. In a lead-acid battery a small current will continue to flow even after one or more cells are fully charged (the main effect of this current is decomposition of water into hydrogen and oxygen). This current helps to fully charge other cells that are lagging behind, thus equalizing the charge state of all cells.

In comparison, the current through an LFP cell when fully charged is nearly zero, therefore the lagging cells will not be fully charged.  Unlike Lead-acid, the cells that are fully charged will raise in voltage extremely quickly, which can lead to instant degradation of the battery if left unmanaged.  Over time the differences between cells may become so extreme that, even though the overall battery voltage is within limits, some cells will fail due to over- or under-voltage, so cell balancing and monitoring is absolutely required.