Your “neck of the woods” refers to the area you come from or where you live currently. While this phrase might have specifically referred to forest settlements during its inception, it is now used more loosely, referring to urban settings as well as rural ones.
“Stomping grounds,” on the other hand, refers to a place that you spend, or spent, a lot of time. For example, I might walk through the halls of my high school and refer to them as my old “stomping grounds,” even though I haven’t spent time there in years.
You might also hear “stamping grounds.” This is also correct! “Stomp” is the modern version of “stampen.” In fact, “to stamp” means to press down heavily on something with your foot. Knowing this, it’s easy to see that “stomping,” or “stamping,” “grounds” means somewhere that you spend a lot of time, literally stomping around or figuratively just “hanging out!”*
Stomping / Stamping Grounds – One of the earliest appearances (in print) of this phrase is in OED: 1862 Harper’s Mag. In this context, you see the original meaning of the phrase: “June 34/1, I found myself near one of these stamping grounds, and a simultaneous roar from five hundred infuriated animals gave notice of my danger.”
Stamping grounds…. A place where animals, normally cattle, habitually gathered.**
In this neck of the woods
The word “neck” has been used before to describe a thin or narrow stretch of land, beginning in America in the first half of the seventeenth century. “Neck of the woods” then evolved to mean “a woodland settlement” by the nineteenth century. The phrase did not have as broad of a meaning as we know it, until the first half of the twentieth century.