"Public prayer is more formal prayer.
It has a formula in which you express those notions that you have in your heart, not in a spontaneous way, but through a structure.
Informal prayer is essentially private.
It is spontaneous: one wants to say what one thinks, especially in times of war, famine, illness, where one is unprepared to say what one wants to say.
For formal prayer, the Siddur (Jewish prayer book) is the standard and most popular repository for Jewish prayer.
But there are also other sources of prayers such as the Mishnah and Kabbalah.
And new prayers are being written today.
Theoretically, there shouldn't be a big difference between the two forms of prayer.
Let's say you are in love and want to propose.
You can propose in any way you want according to the personal relationship.
Sometimes you are tongue-tied or don't know the right way to express yourself.
So, you learn some formula.
The intention is the same but the structure is something already made.
Basically, public prayer represents praying in a community, as a family.
The Siddur is not a collection of private prayers but an attempt to create national prayer.
It has different details, some of which would be pertinent to me now; and some of which would simply be details.
For example, we have a blessing for full health.
In times of illness, this prayer will be very emotional.
Other times, it won't be something that is bothering you.
But I want to express these prayers in a way that reflects the whole nation in prayer."
From "The Art of Preparation for Prayer," a conversation with Rabbi Adin Steinsaltz