While asking this question : Requirements to drive to Mexico from USA I realized I would be better off asking this question first. While driving in personal truck, what would be the closest Mexican city/town/suburb from Texas border that one can visit to experience traditional Mexican culture and food? It would be nice if the city would have a few places to visit. We are not looking for anything fancy. Any places of archaeological importance or places with scenic beauty (just visually appealing).
There isn't a lot, culturally speaking, on either side of the U.S./Mexican border. So where to go depends a lot on exactly what you're looking for.
The main U.S./Mexican border crossings along the Texas border are:
There are other crossings, but they are at essentially unpopulated areas, so for a cultural experience, they aren't really worth considering.
I have never crossed at El Paso / Ciudad Juarez, but Juarez has recently (in the last 5 years) been ranked as the most dangerous city in the world by some metric. I'm not sure I would recommend visiting there, unless you have another reason to go. It's also probably out of your way, depending on what part of Texas you are in.
This leaves the three other Mexican border towns: Nuevo Laredo, Reynosa, and Matamoros.
Laredo and Reynosa are pretty dull. I spent 3 days in Reynosa a month ago, because I have a friend there. But we just spent time together... there's not much to see. I have also crossed the border at Nuevo Laredo countless times. Also boring.
If you want to visit a border town, Matamoros is probably your best bet. It has been 15 years since I was there, but it's close to the beach, and it (like Nuevo Laredo and Reynosa) offers authentic Mexican food.
I suggest reading travel guides (linked above) for each city, to make your own determination about where is ideal for you.
One important thing to consider, in light of your previous question: If you're only in the border regions (within 20km of the border, if memory serves) you do not need a vehicle permit, or even a tourist permit. You only need these if you travel outside of the "Free trade zone".
If you are willing to travel outside of the free trade zone, then consider Monterrey, Nuevo León as a destination. It's about a 2-hour drive from either Nuevo Laredo or Reynosa, and is much larger than any of the border towns. It's also prettier, and has a lot more to offer, both in the city, and around the city (if you're interested in natural sights, etc).
The city of Chihuahua, Chihuahua might be another possibility, but I've never been there, so can't offer any personal suggestions. It's also probably farther away from you, as it's closer to Ciudad Juarez.