Bitcoin Core helps to support the network, whether you open port 8333 or not. It still helps to enforce the network rules.

Bitcoin Core itself is a trustless client which does not trust any other person other than itself. If a miner mines an invalid block which does not follow the network rules, Bitcoin Core will reject it and SPV clients would happily accept it till a longer chain emerges. Bitcoin Core has various tools for developers, the RPC interface etc. It's great for payment system.

Bitcoin Core takes a huge amount of memory space. It keeps a copy of the entire blockchain which is more than 55GB at the moment. It is projected to grow even more if the block size increase. This also means that you need to have lots of free space for it. I don't run it since I only have a 128GB built in SSD. Bitcoin Core misses out on features such as hierarchical deterministic seeds. HD wallets have a seed which the address generates from. This only require the users to make one and only backup in their lifetime (of course you need several copies). With Bitcoin Core, you need to backup every 100 transactions or so since it pregenerates 100 address everytime and uses a change address for every transaction.