A presenter says let me segway into the budget section, and half the room spells it mentally as the scooter. The two words sound identical, which is the entire problem, and one of them is almost never the word you mean.
Two words, one sound

Segue means to move smoothly from one topic or song to another, while Segway is the brand name of a two-wheeled personal transporter.
Merriam-Webster treats them as a commonly confused pair. They are homophones: both sound like SEG-way. Segue came from Italian musical usage meaning it follows; Segway was coined for the scooter, playing on the same sound.
Why writing segway for segue is an error
Writing segway for the transition word is a spelling error despite its frequency, and editors flag it in formal documents.
The scooter's fame pushed the misspelling into casual writing, and Merriam-Webster's usage page notes segue is occasionally found spelled as segway even in edited prose, where it remains nonstandard. In a work email or an essay, the brand spelling of a transition word reads the way a misplaced logo would.
A five-second spelling check
Before sending, ask whether the sentence is about moving between topics; if yes, write segue, because the scooter spelling never carries that meaning.
Three practice sentences: the DJ segued into the chorus (transition, segue); the tour group rode Segways (scooters, capital S); the meeting segued from introductions to targets (transition, segue). Auto-correct often suggests the brand spelling, so the check must be manual.
How segue behaves in sentences
Segue works as a verb with into, as in the talk segued into budget questions, and as a noun for the transition itself.
It borrows from music vocabulary, so it suits performances, playlists and discussions. Awkward segue names the clumsy version of the same move.
FAQ
Is segway ever an accepted spelling of the transition word?
It appears in casual writing but remains nonstandard; formal writing uses segue.
Can segue be a noun?
Yes, as in that was a smooth segue between the two topics.
Do the two words share an origin?
No; segue came from Italian musical usage while Segway was coined for the scooter brand.