I have several questions that all relate to how wanderer and a specific set of artisan's tool play out. The tool sets are cartographer's tools, woodcarver's tools and leatherworker's tools. The most obvious, and the source of these questions, is cartographer's tools since the way that the wanderer feat is worded. The other two somewhat rely on scavenging/foraging, so similar questions arise but aren't as pressing since they don't overlap with wanderer in the way that cartographer's tools do.

What is the difference between having the wanderer (background) feat and having proficiency in cartographer's tools?

Can a PC with Wanderer draw maps? If yes, can that PC draw maps as well as a PC with proficiency in cartographer's tools? If yes, what is the benefit of taking proficiency in the tools over acquiring the feat?

Can a PC with proficiency in cartographer's tools find food and fresh water? If yes, can that PC find food and fresh water as well as a PC with the wanderer feat? If yes, what is the benefit of acquiring the feat over taking proficiency in the tools?

Does a PC with both the wanderer feat and proficiency in cartographer's tools get any additional benefits that they would/could not gain if they only had one? Is there any specific advantage to the synergy of using both?

This same set of questions, particularly the last question, can be asked of woodcarver's tools and leatherworker's tools. And as I'm writing this out, several other sets of artisan's tools such as mason's tools, if the PC is "foraging" for stones, or carpenter's tools, for the same reason but wood instead of stones, or navigator's tools, can be inserted into the last question.

Basically, how do you handle, probably as a DM or as a player during player creation, a PC with overlaps of a knowledge base? Wanderer and and cartographer's tools being the most obvious for me

For ease of reference, Wanderer is a feature from the "Outlander" background, not a feat (unless you're talking about some Homebrew feat?)

Feature: Wanderer

You have an excellent memory for maps and geography, and you can always recall the general layout of terrain, settlements, and other features around you. In addition, you can find food and fresh water for yourself and up to five other people each day, provided that the land offers berries, small game, water, and so forth.

Cartographer's Tools, Leatherworker's Tools, Woodcarver's Tools, and indeed all other "Artisan's Tools" share a single description in the PHB:

Artisan's Tools. These special tools include the items needed to pursue a craft or trade. The table shows examples of the most common types of tools, each providing items related to a single craft. Proficiency with a set of artisan's tools lets you add your proficiency bonus to any ability checks you make using the tools in your craft. Each type of artisan's tools requires a separate proficiency.

In Xanathar's, the various tools are really broken out and described in much greater detail. Assuming that you're using those optional rules when asking these questions...

1. Can a Wanderer draw maps (without cartographer's tools proficiency)? Recalling/reading maps is very different from drawing them by hand, but at the very least it probably helps when compared against someone else without Wanderer. Anyone can try to draw a map, folks with cartographer's tools will be better add it (add their proficiency bonus)... if it were me, I'd probably give a Wanderer advantage on the roll either way. 
2. Can a cartographer (without wanderer) find food and fresh water? If it's marked on a map and they're trying to find it that way, they might be able to read it easier than a non-cartographer would (finding a waypoint on a map is a DC 15 check as suggested in Xanathar's), since they can add their proficiency bonus to the ability check. But otherwise, nothing about the tools suggest they come with other survival-esque field skills, other than that Xanathar's suggests you'll also generally find it easier to guess which direction to head to find civilization (advantage on that Survival check?). 
3. Does a Wanderer + Cartographer's Tools proficient character get any stacking bonuses? See #1 above, tool proficiency allows you to add your proficiency bonus to attribute checks, while I'd probably treat Wanderer as not providing a flat bonus but rather giving you advantage on ability checks that it's relevant to, as well as simply allowing you to automatically succeed on finding food like described. 

Won't go into the other tools, but in general my answers would be similar. 

Thank you. Even the language of your post is helpful, ie stacking instead of synergy.