An offer with deposit has a pre-authorised holding deposit. In other words, you're saving your card information in advance. Upon offer acceptance, the holding deposit will be withdrawn from your card and paid automatically. This takes the listing off the market right away and reserves the property for you. You will not be charged if your offer is not accepted.
As a bonus, owners tend to gravitate to offers with deposits as it shows you are committed to the property. You've pre-authorised the deposit, which shows you are financially committed.
Example: Another renter has made an offer without deposit on 7 Thalia House E14 9RU and it has been accepted. They are given 24 hours to pay the holding deposit and decide to pay it in a couple of hours when they are free. During this time, 7 Thalia House E14 9RU stays on the market.
You stumble upon this property and are interested. You decide to make an offer with deposit, which also gets accepted. Upon acceptance, the holding deposit is paid immediately and automatically. You have secured the home. While the other tenant's offer was accepted first, you paid the holding deposit first.