Easiest way to see difference is to translate tuttavia as although and però as though... Kind of mean the same, but slight difference is still there! Its helps me to understand the difference, as both italian and english are not my native languages, so i use my native language sometimes to dig to very precise definitions of words...
"Ho una cucina, però non cucino" translates to "Ihave a kitchen, but i do not cook". The use of "however" is incorrect in this instance. The sentence would have to be structured differently for the word "however" to be used.
Note the following CORRECT usage: "I have a kitchen. However, I do not cook"