What is the purpose of vermouth in potato leek soup?
Vermouth is a fortified white wine that has been infused with herbs, spices and fruits. You should expect it to add those flavors to your dish. 

Have you tasted the vermouth on its own? Are you sure you can't identify it? If you still think you can't, I'd suggest tasting your soup as you add each ingredient and concentrate on what flavors each ingredient adds.

For what it's worth, Cooks Illustrated has reviewed Martini &amp; Rossi Extra Dry and classified it as "not recommended" due to tasting medicinal and harsh.

Gallo Extra Dry California and Noilly Prat Original French were both recommended, in that order.

They used a pan sauce as well as straight tasting for their reviews. Perhaps you could try that yourself and see what flavors vermouth gives to a pan sauce, as compared to a white wine. You might find out that you're overpowering or masking a flavor in your vermouth with some of your various herbs &amp; spices.