We love thick stocks. It is a friend with benefits. One of the challenges in letterpress printing is working with the "show through" of the impression to the opposite side of the page. When you have a thicker stock it becomes easier to show more impression and have less show through.
These pictures compare 110lb Cover (300gsm) with 220lb Cover (600 gsm) Crane Lettra which is 100% cotton. What you should notice is that they both have a sculptural impression. But the thinner stock on the top does have some show through, while the thicker stock below has even more impression and no showing on the opposite side. That becomes important if you are doing a two sided business card and wish to minimize showing. The depth of impression on letterpress is controled by varying the amount of packing material underneath the sheet being printed. A single sided design is easier to achieve heavy impression because there isn't as much worry about the back of the paper.