I find the linen finish to be more important on cards than on other components. The linen finish allows air to be between cards which affects the way they move against each other (and other things). I don't have a strong opinion on other components. The linen finish makes art harder to see, but it does feel a bit fancy I guess.
I find that certain types of linen finish can cause a significant amount of glare. If the texture is smooth I can move my head to avoid the glare; if the texture is the wrong kind of linen I'll get glare from a much wider range of view angles. For boards and tiles I therefore much prefer matte laminate (as in the first major run of Pandemic, or in recent prints of Alien Frontiers), although linen occasionally also looks good (the boards in Ticket to Ride comes to mind).
If your components get the linen treatment they are probably laminated, which is good for durability. If they're not linen finished, chances are that they also aren't laminated. This is especially true for cards, and unlaminated cards are very prone to scratching and chipping.
Lastly, I recently received two decks of cards from an on-demand printer. One deck was coated, while the other was laminated with a linen finish laminate. The printing on the laminated cards was slightly fuzzier, as expected, but I also noticed a slight green cast. This could be a one-off thing, but it might also be an issue with that particular laminate. Either way I'd suggest getting some samples before committing to either quality.