In the same way that "your" does not mean the same as "you're"
in the same way that they're does not mean the same as their and there.
And sentences should start with a capital letter. But starting a sentence with "And" isn't great either. Nor is starting a sentence with "But" or "Nor" for that matter.
"However" is my favourite for avoiding the "but" rule
However, the iPhone auto spell commonly misses a capital letter when starting a sentence after a quote.