I've made a custom modpack for me and a few friends to play on.

Server loads up fine, but when I try to join the server, it crashes the client saying "The game crashed whilst ticking entity"

I looked at the crash log and it appears that the baby spider was causing the problem.
Entity Type: primitivemobs:baby_spider (net.daveyx0.primitivemobs.entity.monster.EntityBabySpider)
So I disabled the spider family on both the server and client configs and booted up again.

I get the same crash and error trying to join the server, but this time the error has switched to the grove sprite.
Entity Type: primitivemobs:grovesprite (net.daveyx0.primitivemobs.entity.passive.EntityGroveSprite)

I should point out that the server doesn't get any errors, so I don't think it's server side. Yet everything is fine in singleplayer. I can spawn in baby spiders and grove sprites and I don't crash.

I have the same here the crash: https://paste.dimdev.org/jelufexeje.mccrash

I have the same issue.
I've built my own private pack, single player works fine, nothing wrong there, but in multiplayer, client crashes on ticking entity. Entity type is baby spider, the same as in OP. When I disabled baby spider in config, client crashes on ticking entity of grove sprite. I didn't try disabling grove sprite, so it is possible that other entities crash as well.
Here is the crash log from grove sprite crash crash-2019-02-24_14.39.54-client.txt and from baby spider crash crash-2019-02-24_14.02.36-client.txt

This is the same issue that I posted #219. Disabling The Spider Family and Grove Spirit appear to make the client not crash any more