How relevant is Self Organizing Maps in today's science?
One important and well known (in their respective research domains) usage of SOMs today is the usage of SOMs for data visualization and clustering.

The U-Matrix uses a SOM with a lot of neurons in order to achieve emergent behavior. Based upon the SOM, a so called U-height is calculated for each neuron.

For a neuron $n$, a weight $w_n$ of the neuron $n$, as well as a set $NN(n)$ of immediate neighbours of $n$ the U-height is calculated as:

$$\sum_{m \in NN(n)} d(w_n - w_m)$$

$d$ is the same distance as used in the SOM training. This U-height is used to visualize the SOM and can be capable of catching quite interesting 
inherent structure. It is visualized by using a color scheme similar to topographical maps:



The U-Matrix uses distances for visualization. It can be combined with the so called P-Matrix, which uses densities, to use the density of data points as well, and allows the combination of densities and distances. Based on the U-Matrix one can do classification or clustering as well. An R package is distributed by the research group of Prof. Ultsch.

I wouldn't say that it's important in Machine Learning per se, but in the fields of Data Mining and KDD it is, at least academically. You can clearly see this if you look on the reception of this or similar work on semanticscholar or researchgate.

Disclaimer: I studied with Prof. Ultsch, the head behind U-Matrix and U*-Matrix.

NB: Sorry for the missing links to the papers, I haven't got enough reputation yet to post more than two links.

Papers:


U*-matrix: a Tool to Visualize Clusters in High Dimensional Data, Ultsch 2004
Self-Organizing Neural Networks for Visualisation and Classification, Ultsch 1993


I even spotted the U-Matrix in the wild here. Also see the sompy package.