If the brown grass is an annual crabgrass, it dies each year after going to seed. If that's the case, you need to prevent the crabgrass from growing next year. Use a pre-emergent fertilizer in the spring, timed correctly to prevent the crabgrass from germinating, and/or spray with a selective herbicide that targets crabgrasses.
If the brown grass is dead desirable turf grass species, then you need to dethatch or power rake. Amazon sells electric ones for $100 that will do the job. The rotating tines loosen the dead grass and you can rake it and dispose of it. If the lawn is thin after this raking, you should put down seed. Fall is best for cool season grasses. If you miss the chance to do it this fall, you can do it in the spring, but not if you put down a crabgrass preventer, since it would also prevent the grass seed from establishing.
If the alive desirable turf grass is yellow and not green, you need to fertilize. Generally you'd do spring w/ pre-emergent to prevent weeds/crabgrass from germinating and a fall fertilizer (no need for pre-emergent). These are the best times since cool season grasses grow in the spring and fall.
Both lawns look to be the same color green, only the neighbor's lawn has less dead grass. Other than fertilizing a regular amount on the spring/fall schedule, I wouldn't change anything.