The above 2 answers explain specifics without the underlying reason why PP was required.
In the first one, the size of the conservatory was almost certainly bigger than the %age increase allowed under Permitted Development Rights (PDR). The neighbour with the attached garage has a bigger volume of size of house so would be allowed a bigger structure under PDR.
In the second one, you always need PP if you want to build closer to the public highway than the existing house. So if you live on a corner, extending to the side requires PP. Its nothing to do with whether the road might in future be widened.
So what's the answer? You can use PDR if you do not live in a Conservation Area a National Park or an AONB. You can extend a terraced house by up to 10% of the volume or 50 cubic metres (whichever is the greater). You can extend any other sort of house by 15% (or 70 cubic metres) whichever is the greater. But these volume limits apply to the house as originally built - so if you have already extending using PDR, part of your allowance will have been used up.
Because the rules are complicated, it is easiest to phone the Planning depet at the local council and ask for advice. They are there to help for free.