Tube amp speaker output 4 ohms VS 8 ohms

Are you likely damaged the amp (or speakers) when you connect 4 ohms speaker to 8 ohms tap of the amp ?

No.

Do you get different sound with connecting a 4 ohms speaker to either 4 or 8 ohms tap of the amp.

Yes.

What is the best connection to get the best performance.

Try both and use the one that sounds best to you.

My experience is that there is no way to tell without trying. The sound will be much different in most cases. I have had 4 ohm speakers sound better on 8 ohm taps many times. You certainly will not hurt anything by experimenting. If you run searches in the right places you will find that it is very common for people to experiment.

The 4 ohm taps utilize more transformer windings and can deliver more current which typically equates to better control over the drivers. However, many times this will be at the expense of "sparkle" on the top end and almost always ultimate spl output.

Try both and go with what sounds best without looking back.

8 ohm should be fuller sounding but may not have the bass control of the 4 ohm

Thanks everybody

The combination of your personal preferences, amplifier, speaker, and speaker cables make your question impossible to answer.

Try both taps and use the pair that sound best to you.

"Best performance", as you say, is subjective.