I find that it can be quite difficult to get an accurate comparison when looking to change suppliers, because such comparisons often compare new tariffs with the expensive tariff you will be automatically moved to if you don't do anything when your current contract ends, rather than the tariff that is about to end. For that reason, before choosing a new supplier, I do the maths for myself (based on the same annual usage figure that their or the comparison site's supposed saving is based on) to check that we are arriving at roughly the same amount.
A month ago, I had to do this for both myself and also for my mum (I can think of more enjoyable ways to spend a week!)
Given that wholesale energy prices have risen a lot recently, I was amazed to find my mum a much better deal than the one that was ending, and with a well-known company too (Eon).
Usually, I am quite risk-averse, but in an attempt to keep our own energy bills down, I have taken a risk with our supply, by going for a new company (Outfox the Market) with a different way of selling. It is not fixed-price (and there has already been a price rise only a few weeks in!), but we can leave at any time. Only time will show if it was a good decision or not!