NewRisingSun used IDA to compare the disassemblies of BIOS ROM v1.00.00 and v1.01.00, and found the following differences:
- 6845 initialization routine now clears light pen latch
- FDC access routine at C9E now sets the two drive select lines to value 3 if all motor bits are off
- FDC I/O access routine at EF7 deals differently with unwanted pending bytes in non-DMA mode
- Interrupts are now disabled while setting parallel port strobe high and low.
- F1 during boot sets mono mode
- 1985 copyright date

Gloriouscow also noted the following:
- the main bug fix in 1.01 is disabling DMA extended write mode

...and went into extensive detail at https://forum.vcfed.org/index.php?threads/upgrading-the-tandy-1000-bios-rom-to-v1-01.1256852/