I currently use a V60, but I'm interested in using a Chemex (primarily because of the bigger capacity for guests).
Assuming all variables are correct for the specific brew method (grind size, weight of coffee, filters etc) but the coffee & water used are from the same source.
What would be the differences between the taste of a cup of coffee from a Chemex and a cup from a V60?
Here's a few more things to add to @fredley's answer.
Since the water is going to be in contact with the grounds for a longer time (1- using greater volume of water, and 2- Chemex filter will be slower) you're going to get more of the bitter notes that you get (in general) by using a longer brewing time. Acidity will be offset somewhat by the bitterness. I find Chemex to be like a filter-coffee version of the body of a French Press preparation, though with no sediment. Compared to the throughput of V60 filters, @fredley hit it on the nose.
Though you are asking about taste specifically, there are other factors.
Like @fredley said, the filter paper in Chemex is much thicker and water drains through it significantly more slowly. You'll probably want to use a coarser grind (and very even grind) than you would for V60 or Melitta-style filters. Fines are the enemy of Chemex, causing the filter to clog and welling (see this question about "welling up" in a Chemex for more on that). Certainly use the best burr grinder you can and don't delay in the pouring action.
Technique is "more important" (read as: "finicky") with Chemex, I think, than other pour-over systems. See preparation guides such as from Chemex itself, or Blue Bottle, or Stumptown, or search for others.
So there are two main differences between the Chemex and V60:
The V60 has a much finer filter than the Chemex, so extraction is much faster through the V60 filter, and more oils/solids come through.
The V60 also has ridges, allowing coffee to easily pass through the filter evenly, rather than all having to exit through the bottom - washing out the grounds in the tip of the filter. The control you have over brewing time is much finer with the V60 as a result.
As for the actual effect on taste, this will depend on the coffee you're using, and how you like it. The main quantitative differences will be the amount of oils and solids in the brew, and a longer brewing time overall.
