"Convicted" is a neutral, technical term meaning "a court found him guilty of that crime". "Condemned" can mean the same thing, with the added tone "the crime was particularly heinous or immoral, and he got served with a fitting, just and hard punishment". It includes a moral judgement about the crime and about the deservedness of the punishment. There are also meanings of "condemned" that are used in a religious context (so the same as above, but without the court of law) like "the sinner is condemned to hell for all eternity", "sisyphos was condemned to eternal useless labor". Metaphorically, it can also be used to describe someone without the power to exact punishment just telling off somebody for their immoral behaviour, like "the newspaper article condemned his doings as acts of barbarism".
The general difference is that "convicted" is neutral in tone. "Condemned" includes a particular tone, and religious and moral connotations, which might be unfitting in some cases.
Edit: Take the above with some grain of salt, might be at least incomplete, maybe somewhat wrong. After consulting the internet, I've found out that there are even more meanings and nuances, which I didn't know about. Sorry for being an arrogant non-native-speaker trying to score internet points ;)