Found an error in my published conference paper I submitted a paper to a conference, and got it accepted. While I was preparing for the camera-ready version of it, I made a mistake and an error slipped in.

After the firm deadline, I asked if a correction was possible, but they said it's too late since it had gone through all the procedures. So, it seems my conference paper will be uploaded online with an error, which is not too obvious, but obvious enough that a careful reader can spot it.

I'm writing a journal version of it, and making sure it doesn't contain any error. Would it be okay to have the journal version published error-free, while the conference version has an error?

Many say that no one actually carefully reads conference papers as they are mainly aimed to let others know new results, and say that it is journal papers that others read with more care if available.

This is my first published conference paper, and it is really embarrassing. I was obssessed with finding typos, and blind to technical errors.

Thanks.
The reality is that the best answer is simply "get used to it". It is sad that this happened to you in your first conference paper, but things do get published with mistakes, and there is really nothing very much anyone can do about: you can proof read papers as often as you want, and there will always be mistakes. I'm not saying this because I am a nihilist but simply because I'm pragmatic: of course I want papers to be perfect; it just isn't practical.

The interesting question is only: how severe is the error. You don't answer this in your question above, but if it is really only a typo or misspelling, then it would not be worth your sleep. If it is a mistake that a reasonably educated reader will be able to understand as a mistake, the same would probably apply. If you submitted a proof that has a mistake and upon further thought you realize that the whole theorem is not true, that would be a separate matter -- but you wouldn't introduce such an error while dealing with the final version of the paper.