It's obvious that one of the reasons you should invoice quickly is because you're more likely to get paid faster. Getting paid faster is always a good thing.
However, what many freelancers don't understand is that that first invoice lets new clients determine whether they want to continue to work with you. While I can't cite statistics on this, many new clients start smaller projects with freelancers to get an idea of what it's like to work with them and whether they can afford working with them in the long term.
As an anecdote, my team recently started working with a new videographer. She does superb work and we're quite keen to continue working with her in both larger interactions and to have her as our standing resource when it comes to video production.. We also have a few more projects forecasted) right now.
There's only one problem: because she hasn't billed us after a few weeks, we haven't initiated a conversation with her about the next projects. While we're perfectly happy paying for the total fees for the first project, we're hesitant to start a conversation about the next ones in case the final bill is higher than we anticipated.
So, not only is she missing out on the revenue for the first project, she's missing out on the revenue from the projects we could be working right now. Not to mention that fact that we're not getting the materials we need to move forward with these initiatives.
In other words, delayed billing is a loss for both her as a freelancer and for us as a client. In case you're curious, yes, we will send her a follow-up note in the next few days asking for an invoice so we can pay her.
Of course, contracts and agreements help with this considerably. But agreed-to costs can sometimes shift one way or the other. That's why that first bill is the true gauge many clients use to make future sourcing decisions.
The cleanest solution is to bill as soon as the project is complete - unless you have very, very good reasons not to do so. (For instance, say you know the client's mother just passed away. You obviously don't want to send an awkward "Sorry for your loss , payment instructions enclosed" email or invoice.)
From your perspective, sending an invoice quickly may seem like sending someone bad news in a hurry. That's reasonable, as many people don't like paying bills. However, your best clients are the ones who are happy to work with you, pay you for the value you deliver, and hire you for the next project.
They just want (and need) to know how much it's going to cost them.