Though/however

She's good. I'm better though.
She's good. But I'm better.

I always use 'though' like 'but' as shown. Loads of people say that 'though' at the end of a sentence has the same meaning as 'however'. Why is that? Is it 'but' or 'however'?

Can I use 'however' instead of 'though' at the end of a sentence? Is that formal?

________

I use 'however' at the beginning of a sentence when I want to emphasize that the sentence has no relevance to the previous mentioned sentence. As in:

You're very bad at this. However, keep training and you'll get better eventually.

How should I use 'however' at the beginning of a sentence?

Thanks.