You've got the general form down, but it's (as it almost always is) how the details are arranged. Looking at the example map, the initial thing that jumps out at me is that the example is literally covered with labels. The labels are very approximately uniformly distributed across the whole map surface and the sheer quantity of text affects the appearance of underlying terrain, adding a texture that's not specifically part of the terrain, but affects the experience of the terrain. It's one of the reasons why running labels along features like valleys and roads and rivers can be so very useful: it emphasizes the feature without changing the direct presentation of the feature.
My initial experience of your map was that it's from a 1990s school atlas done for a specific country that doesn't like its neighbors much. It feels very sterile compared to the example because it's lacking details like the map frame (outside the neatline and grid) that would tie it to the page as well as lacking the subtle terrain texturing shown in your example. That sort of minimalism and hyperfocus screams 1990s to me.
The texture on the 1950s map is in large part due to the offset printing process. It's the same as with the lines. It looks like that plate was printed with an offset printing process (possibly simple CMYK, possibly a couple of process colors) and that technology requires some compromises that you don't get with more modern technology like dye sublimation. For example, it's hard to get very thin lines on uncoated paper, so the non-coastline areas are actually chains of dots. It's impossible to get solidly uniform areas of colors unless you use one of the primary inks, so dots are used instead and density plus overlap of dots composed of different inks combines with the page background color to give the final visual shade. Generally, the printer wants to avoid putting too many dots in the same place, so that type of screen process used to make the printing plates can have a big impact on the final experience of the map. The kind of paper stock used for the printing process makes a big difference as well: fully-coated stock keeps the individual dots nice and small (allowing smaller dots and higher effective resolution), while cheaper stock will let the dots spread, limiting the maximum effective resolution.

I like to define a map as artwork showing an abstraction of a particular place done for a particular client for a particular reason and in a particular medium. Yes, I'm very particular about that definition. Some folks like to produce a picture of a map, where the map is part of a larger composition such as the scanned page in your example map above. In your example map, the medium and even the context plays a big role in the final experience of the map. You're working digitally, so you have options that the 1950s folks didn't have and it shows in the final results. One of the hardest parts of making retro maps is recognizing the constraints of the media used by the original map producers (clients + artists) and adhering to those same constraints.