Yes it makes a difference, at least on mine.
On transmit narrow mode is 2.5 kHz deviation and wide mode is 5 kHz. I can easily hear and see the difference when sending DTMF, 1750 Hz and CTCSS tones to a receiver with a scope or voltmeter on the audio output.
Go to VFO mode on a vacant frequency and open the squelch. Then switch between wide and narrow. You should be able to hear the audio level of the white noise get louder when set to narrow. Something changes in the receiver between wide and narrow but I can't tell if it really changes the bandwidth or just the audio level.
Here is a review here where they say only the receive audio is louder on narrow but the selectivity doesn't change:
Baofeng UV-5R review (UV-5RE) | PE1RQM
Since the UV-5R is a SDR there is no IF and no IF filters. Receive bandwidth is done via software. Some functions and features could possibly change in different firmware versions.