Apart from what Andy mentioned, there is another difference which could be important - write-host directly writes to the host and return nothing, meaning that you can't redirect the output, e.g., to a file.
---- script a.ps1 ----
write-host "hello"
Now run in PowerShell:
PS> .\a.ps1 > someFile.txt
hello
PS> type someFile.txt
PS>
As seen, you can't redirect them into a file. This maybe surprising for someone who are not careful.
But if switched to use write-output instead, you'll get redirection working as expected.