Aside from not having to download the entire bitcoin blockchain when using electrum, what other advantages are there over using Bitcoin core 0.17.1?
Also if I chose to use my own electrum server, what advantage is that over Bitcoin Core 0.17.1?
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Using Electrum has some advantages and disadvantages over Bitcoin Core. Electrum is a bit more feature rich than Bitcoin Core. It supports hardware wallets, 2 Factor Authentication with a third party paid service, (better) multisig, and mnemonic phrases. There's probably a few other things that I've forgotten. However Electrum comes at the risk of decreased privacy and security. You are sending all of your data an Electrum server. That server knows all of your addresses and your transactions. That server can also withhold transaction information or refuse to relay transactions that you make. Electrum servers can also currently push malicious phishing messages to your Electrum wallet. That issue is currently in the process of being fixed (it requires a protocol change).
If you use your own server, there are a lot of benefits. If you trust that the server software is not malicious, then you won't have the cons mentioned above. Your transactions will be private and the server won't be withholding or refusing to relay transactions because you control it. Since it is your own server, you can have Electrum installed on multiple devices pointed to that server too. However running your own server has a cost. It requires a fully synced bitcoind with transaction index enabled. The electrum server will also maintain its own indexes. Overall, the setup can use over 250 GB of disk space, possibly even 300 GB. It also takes a long time for it to sync and build all of the necessary databases. Running an electrum server also uses significant CPU and RAM especially during the initial setup.
If having the full blockchain with additional databases is too much, you could instead use Electrum Personal Server. This is like running your own full blown Electrum server but only for your addresses and transactions. It can run without transaction index and be ran with a pruned bitcoind so you don't need to store the whole blockchain. With some manual configuring, you can also use Electrum Personal Server with Electrum wallets on multiple devices.