This is what I'm thinking too. In every smart thermostat I've ever purchased, there was a jumper factory preinstalled between RH and RC. Removing that jumper when it's required can result in the thermostat being unable to send heating and/or cooling calls.Check the install instructions closely on your new thermostat, some of the newer ones require a jumper wire, if you wired it the same as your old one that could be an issue.
I had to run new 18/5 thermostat cable on one of mine, and meter out the cables on the other (the aforementioned expected color set was ignored by the original installer). Pretty straightforward operation, except the upstairs air handler did not have a dedicated C wire connection, so I ended up wiring it directly on to the ground screw of the secondary side of the transformer. Hasn't given me a bit of trouble.We had to correct some things when we installed our Nest thermostats (two different units). It wasn't working when we first installed it, finally had to call and they walked us through it to correct the wires.