With a pruning client you only need to store a small subset of the blockchain and there are no major changes in regards to trust. Your client still verifies every single transaction, but it removes data from the chain that it won't need anymore in the future (e.g., an old transaction that already has been spent).
With a SPV client your client only stores the chain headers and then only requests the required data from a server. SPV clients need a small amount of resources and can run on mobile phones. At the same time most of the important functionality (transaction signing, transaction verification, ...) happens still locally and the level of trust that you need to put into the servers is quite small.