I currently have a Philips Hue Bridge and a Bosch Smart Home Controller. Both bridges have devices connected via Zigbee and I can control these devices with Home Assistant via their API. My question is, what are the advantages and disadvantages of connecting Zigbee devices directly to Home Assistant via Home Assistant SkyConnect? I have connected lamps, sockets and buttons to my Phillips Hue Bridge. I have connected sockets, radiator thermostat and room thermostat to the BSHC. I have imported all devices into HomeKit. This would also be possible via HA. But would it also be possible, for example, for the room thermostat and radiator thermostat to be connected to each other as before and for me to be able to control the radiator thermostat with the room thermostat?
The benefits of Zigbee2MQTT is that it is the leading open zigbee system out there now. Hue, is a great ‘walled garden’, if you can limit yourself to only Hue and Friends of Hue products.
IMHO, the best way to go is to initially keep your HUE bridge and then add via a Docker setup OUTSIDE of Home Assistant one or better two (one for production and on for testing) Zigbee2MQTT systems connect to Home Assistant via MQTT. While the ZHA system build into Home Assistant seem appealing, the amount of development and support effort put into it pales compared to Zigbee2MQTT, gotta pick your battles…
There are a number of ways to ‘interconnect’ all of these systems, HA, Hue, Zigbee2MQTT (and I am guessing, due to no experience, Bosch). So there is really no ‘right’ answer. Just shoot for the most stable combo that has the most KISS setup and operation.
I started with Hue the day it was released and have been pleased with my use of it. That said, I have one Hue hub connect bulb left in my ‘system’ today, every other Hue and other manufacture device running in Zigbee2MQTT.
To your specific use case:
But would it also be possible, for example, for the room thermostat and radiator thermostat to be connected to each other as before and for me to be able to control the radiator thermostat with the room thermostat?
Unfortunately, it depends on a lot more specific details about the devices than you have shared. Are they all zigbee? In my recommended path, are they all supported by Zigbee2MQTT?, then even then are all the devices you specify able to zigbee ‘bind’ with each other?
No one said this was ‘easy’, that is why there is 
and 
Good hunting!
zigbee2mqtt won’t try to force you to upload your data to cloud or try to milk some subscription money out of you, like philips hue: Philips Hue will force users to upload their data to Hue cloud