Square Cash is often described as a competitor and imitator of Venmo, the popular app owned by PayPal that makes it possible for friends to send one another payments by smartphone.

In recent months, though, Square Cash has quietly bypassed Venmo to become the most frequently downloaded financial app of any kind on both Apple and Android phones, according to Apptopia. At various points over the past week, Square Cash was ahead of Venmo and Twitter on the charts of the most downloaded iPhone apps.

The big banks had aimed to challenge Venmo with their own service, Zelle, but that has not posed a credible challenge to either Venmo or Square Cash so far, the rankings show.

Mr. Dorsey and his team made several early design choices with Square Cash that helped make it different from Venmo. As would be expected from Mr. Dorsey, Square’s app has a clean, green interface, compared with Venmo’s busy blue dashboard.

Square also made an apparently boring technical decision, to use the debit card networks rather than bank transfers, to move money around. That has made it easier for Square Cash to put money instantly into the bank accounts of its users, and to collect a fee for the service.

These differences appear to have made Square Cash more popular with lower-income customers who more often need instant access to their money and who don’t have as wide a variety of credit cards and other financial options.

“We are reaching an audience that may not have a bank account or may not have a full suite of services from a bank,” Mr. Dorsey told analysts last quarter.