When the liver produces ketone energy (acetoacetate), it results in the creation of breath ketones (acetone), which indicates the production of ketones. Conversely, blood ketones signify the incapability of your cells to use stored ketones for energy.

If you're concerned about your body's incapacity to utilize ketone energy (such as in the case of type 1 diabetes), a blood ketone meter would be appropriate.

On the other hand, if you want to determine how your body metabolizes fat into ketone energy, you should opt for a breath ketone analyzer.

Why Breath Ketones Indicate Ketosis Better than Other Methods.

When fat is metabolized into ketone energy (acetoacetate), breath ketones are released (acetone). The more fat that is used to create ketones, the more breath ketones are released.

When your cells need ketone energy, all acetoacetate is used (except for what was turned into acetone). Neither urine or blood ketones are formed in this context.

When your cell energy need decrease, the excess acetoacetate have two fates: 1) when cells are not adapted, it leaves body in the urine. 2) when cells are adapted, some acetoacetate could be saved for later, when ketone energy is needed again.

In other words, when there is any breakdown of fat into ketone energy, breath ketones are released. When the need for ketone energy decrease, the excess ketone molecules will be disposed in urine or packaged into a more stable molecule (beta-hydroxybutyrate) which can circulate in blood until ketone energy is needed again. Urine and Blood Ketones are present when ketosis decrease. 

For healthy people, measuring blood ketones does not mean much, it could be nothing and you still be ini ketosis but your cells need all ketone energy it can get. It could be high from accumulating over a period or just zero because your have not been in Ketosis in a while.

Blood Ketones are only a valid measure for people with type 1 diabetes or type 2 taking SGLT2 inhibitors (also called gliflozins) medicatio, as the Blood ketones can not be converted back to acetoacetate when blood glucose is high.

They have a vague correlation - the higher breath ketone levels measured, the higher the probability is of finding blood ketones.

Breath ketones correlate to ketogenesis (stored/digested fat that turns into ketones). The higher breath ketones, the more fat is metabolized in the liver.

Phase 1: When energy needs increase, and there is a limited amount of carbohydrates to use for energy, ketogenesis starts and make ketone energy. In this phase, both the primary ketone AcetoAcetate and the blood ketones (beta-hydroxybutyrate, if present in the blood) is consumed. Breath ketones are proportional to the AcetoAcetate created. No blood ketones are formed in this phase. View animation

Phase 2: When energy needs decrease, the liver is still producing AcetoAcetate, and cells have enough energy. The cells are intelligent and can "repackage" the AcetoAcetate into the storage form beta-hydroxybutyrate (blood ketone) and leaves it into the bloodstream for later use when energy needs increases again. Breath ketones are still proportional to the AcetoAcetate created and decrease along with the production of AcetoAcetate. The blood ketones signal that energy needs are decreasing and the liver will produce less AcetoAcetate. View animation

So when energy need increase (and carb availability is low) ketones are made and used (if you had any blood ketones, they would be used to and level decrease). Ketosis increase and breath ketones rise.

When energy needs decrease, the cell's save excess ketone energy into blood ketones. The blood ketones signals decreasing the need for energy and ketosis also decrease (so will also your breath ketones).

The level of blood ketones reflects how much the cells saved when the energy needs decreased, not the level of ketosis. Comparing levels of breath and blood ketones is therefore complicated as they are not formed under the same circumstances.