The pro

• reporting one account for taxes is easier than multiple
• It is easier to gauge your performance and spare cash if there is only one place to check

The cons

• A single point of failure can be problematic. If your broker goes bankrupt the shares are still yours. But it may take some time to claim ownership and transfer them to another broker while you are still exposed to market risk. (And remember, bankrupcty rarely happens in the good times)
• As an extension of the first bullet point, some (neo-)brokers are not very profitable. It is not guaranteed that they will be around for long. For example, the German counterpart of Robin Hood is estimated to run on 16 mio Euro a year (Source, German only)
• depending on jurisdiction capital gains by shares might be taxed differently depending on when they were bought. Keeping two accounts helps to tax them correctly
• one might have different risk profiles for different accounts. Let's say one account that runs long term ETF plans and another one that includes stock picking for the fun of it. Separating these allows to limit gambling to one dedicated account
• different brokers have different cost structures. This seems to be less of a problem in the US but here in Germany standard provisions for an ETF plan are 1.5-2% range and only some ETFs will be on provision free plans. Having multiple accounts can help you to minimize provisions