“I wanted to go to Wendy’s; she wanted Pizza Hut.”
Both of these can exist as separate sentences and it would be grammatically correct to write them as such. But you can link them using a semi-colon so that when they’re read outloud, there’s less of a pause like there would be if there’s two sentences.
Alternatively, they can be used in place of commas when writing a sentence containing a list that might get confusing if you just relied on commas to separate the list items:
“My role models are George Washington, the first president of the United States; Nikolai Tesla, a famous inventor; and my Uncle Stu, who beat cancer.”
If I had just used commas to separate the list items and the sub clauses, it could be read as though George Washington and “the first president of the United States” were separate items in the list.