There is a noticeable difference in handling, stability and lap time in my opinion. Lap times are usually 1-2 seconds quicker offline.
Recent example: Last week's
Car Of The Week, the 1963 Corvette Coupe C2 Race Car, where there was a Time Trial at Willow Springs (Big Willow).
Offline with Sports Soft tyres equipped, you could take both the long second turn and long final turn with your foot pinned to the floor. Trying to do the same online with Racing Hard tyres would cause the outside rear wheel to spin up and the vehicle to violently snap with little notice. Over all lap times between the two where almost the same, even though online tyres were supposed to have more grip and produce faster lap times.
It's good practice to tune your cars in an online room rather than offline if you intend to do some competitive online racing. What may be a stable setup/tune offline often becomes a lot more edgy online.