tl;dr: Permanent filters win on cost over time. Paper filters win on performance, clean-up, and health. Flavour is personal preference. Composting paper filters and spent grounds alleviates most of the environmental benefit of permanent filter. My conclusion: paper filters, discarded into a compost pile, win hands down.
Use of permanent versus one-time-use filters is a perennial question, with both concrete and personal-preference arguments. In addition to the link in the original post, there are a few other online articles with comparisons, such as:
From those and my personal experience, I extracted an overview and conclusions, below. Unfortunately I've written a lot of text here, but there's a lot of facets. I've focused basically on paper versus metal mesh, as the most common types; I maintain the conclusions are similar regardless of material.
Material. There are several materials used for permanent filters, including the following:
- metal mesh (stainless steel or "gold tone");
- nylon (or other plastic) mesh;
- perforated metal (thin metal sheet with holes);
- fabric (e.g., flannel, muslin, hemp).
All of these surely have different properties, but properties that I focus on here are common to most.
One-time-use filters are predominantly paper, but could also be bamboo, natural fibre, or other material. See also this question for a comparison of brown versus white paper filters.
Performance. Paper filters are a finer filter; a permanent filter will permit some larger particles through to the cup. I find that a pot brewed with a permanent mesh filter will have a fine silt of coffee grounds at the bottom. For this same reason, some recommend to use a coarser grind for permanent filters.
Paper filters are also claimed to filter out some compounds (e.g., diterpenes such as cafestol, which may increase blood cholesterol levels). Coffee brewed by other (non-paper-filter) methods have a higher concentration of cafestol in the resulting beverage. See article  for more health claims of paper filters. On the other hand  says that paper filters filter out some good things, such as antioxidants, but I haven't seen that claim anywhere else.
Flavour. Link number  claims better taste on paper filter than permanent filter (I happen to agree). Permanent filters should also allow more oils to pass through, which will certainly have an impact on flavour; this is probably personal preference. The grit that passes through will also have a bittering effect over time, but probably not significant if you're drinking right away. Some (e.g., ) say nylon/plastic mesh will retain off-flavours over time.
Cost. Varies greatly, but one can find permanent filters that are said to last for "years" for a few quid ($/€/£) up to 20-30 quid. Though dependent on usage and other factors, you'll have reasonable expectation of saving money over time by using a permanent filter.
Clean-up. Clearly, the spent grounds from re-usable filters must be cleaned out; paper filters and their spent grounds can be pitched wholesale. See another question about how often and how much to clean a permanent filter. On the other hand,  claims that permanent filters are cleaner, but I don't agree.
Waste / environmental issues. I couldn't find any data on how much energy is used to produce a paper filter versus a reusable filter, but many uses would certainly be necessary to break even! As far as waste volume though, you'll make up the difference with paper filters pretty quickly (but composting the paper instead alleviates this!). Cleaning permanent filters additionally requires water and detergent. Though, you may wish to rinse your paper filters, so water use is debatable. Ultimately, reusable filters will eventually wear out and need to be replaced.
My Conclusion. For my money, personally, I prefer the paper filter:
- Better flavour and texture (less bitter, less grit);
- Convenience (easier to use and keep clean);
- I compost the whole thing (paper filter, grounds and all), so the environmental impact is basically on the production end, which I'm okay with.
- Marginal health concerns (cafestol/cholesterol; avoiding plastic whenever possible).