And how does a full node (which isn't mining) benefit the network?
This is a bit like asking if a single vote matters in an election where there are thousands (or millions) of voters. It can seem a bit like that one vote isn't really going to matter much at all, however if EVERY participant thought that way, then nobody would vote and that single vote would be ALL that matters. As such, it's in the interest of the entire system for everyone to act as if their small contribution really does matter (because it's only when they do so that they can make sure that no single participant has excessive control).
Here are a few ways that a non-mining node benefits the network:
1. It enforces all consensus rules on all transactions and blocks that it receives and forwards.
2. It provides a source of pieces of the blockchain for new nodes which wish to join the network.
3. It provides a list of well-behaved nodes for other nodes to connect to.
4. It increases interconnectivity, reducing the risk that the network becomes fractured.
Can bitcoin accomplish these things if YOU don't run a node? Probably. But, if too many people feel that way, then any or all of these begin to break down. Simply to exist, bitcoin NEEDS some threshold number of participants to perform these actions. As additional nodes are added above that threshold, it improves the durability of Bitcoin and reduces the power that any other participant has.
If you feel you are benefitting from the very existence of bitcoin, then you may want to consider doing your part to ensure its continued existence. You get to decide what "doing your part" means. This is a voluntary system and nobody is forcing you to participate in any particular manner. It's only your own sense of responsibility and community that can motivate you. If running a node feels too expensive to justify for the amount of benefit you're receiving from bitcoin's existence, then there are other ways you can contribute to the community. Some great ways to give back are:
1. Encourage others to use bitcoin when they are going to pay you for some reason.
2. Encourage others to accept bitcoin when you need to pay them for some reason.
3. Learn more about how bitcoin works, and what its benefits are.
4. Participate in discussions (both online and in your daily life) about bitcoin and educate others about how bitcoin works and what its benefits are.
5. Run a pruned node
6. Learn to program, and write code or test cases for Bitcoin Core.
7. Support those politicians that are not hostile to the concept of Bitcoin.