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There's a couple of things - one is efficiency of running the home eg damp walls conduct more heat to the outside, the other is how temperature, humidity and damp (condensation) are connected. These affect ambient temps and humidity, which are really important. Thermal comfort (how you feel) is also influenced by other things - a really significant one is radiant heat - if the surfaces in your line of sight (your body's thermal line of sight, really) are cold, you will feel colder. If they are warm you will feel warmer. Internal insulation provides a warm surface. It also improves overall thermal efficiency (less heat lost to the outside), so whatever heat you are inputting leads to warmer temps. The big thing to watch out for is interstitial condensation between the warm insulation and the cold wall. The existing cavity will still be doing a job to prevent damp from the outside getting to the inner leaf, and it will still be doing a job of reducing heat loss but much less significant compared with the new insulation.
Edit to add: In the summer, another thing to consider is that internal insulation covers up your thermal mass, so overheating is more likely to be a problem - not because of the insulation but because you lose that exposed thermal mass which acts to buffer heat gains.
hoefla-
Edit to add: In the summer, another thing to consider is that internal insulation covers up your thermal mass, so overheating is more likely to be a problem
How does that work? Presumably overheating is due to too much heat moving from outside to inside. That process would be slowed by internal insulation. It might stop temperatures equilibrating overnight (assuming that the outside temperature drops below that of your house), but I'd suspect it's more important not to allow your house to get too hot during the daytime in the first place.
ffm
This is really interesting, since a common recommendation is to fit internal insulation. For example, this is being recommended to us because the cavity in our walls is pretty nominal (~50mm). The downside is that the inner leaf of blockwork then doesn't warm up so the dewpoint moves inwards. I've read people on here saying how good they felt that interior insulation is, but I'm interested in whether this effect can cause an issue and reduce the benefit of the existing cavity.