Summary: Fairtrade International gives priority to social problems (alleviating poverty and good working conditions), Rainforest Alliance to environmental aspects (protecting natural resources) and UTZ is somewhere in between. There are indications that Fairtrade International is the most strict label and Rainforest Alliance the least, but all 3 labels have their strengths and weaknesses.
UTZ, Rainforest Alliance and Fairtrade have a lot in common. All 3 organizations try to transform agricultural production and trade systems, improve the working and living conditions of farmers, forbid child labor, and protect biodiversity and ecosystems. The 3 organizations cooperate with each other to achieve this, but they also said that they believe each label has its own value.
The main difference between the 3 organizations is which goals they feel are most important and how they try to achieve those goals. Roughly speaking Fairtrade International gives priority to social problems (alleviating poverty and good working conditions), Rainforest Alliance to environmental aspects (protecting natural resources like wildlife and forests) and UTZ is somewhere in between.
You can see this more clearly in how each label does business.
Fairtrade International (FI)
FI gives farmers a minimum trade price and premium which are set in advance. This gives farmers more income security, but the downside is that it reduces the incentive to keep the quality of the products up (although there is still an incentive to sell high quality products at a higher price). People have criticized the fair-trade model for being unrealistic with too high prices and consequently not being sustainable nor applicable to all farmers world-wide.
Rainforest Alliance (RA)
RA does not offer minimum prices or mandatory premiums. Farmers have to meet a list of sustainable agriculture principles like conserving local wildlife, forests and water resources, minimize soil erosion and treat workers fairly.
Unlike FI, RA doesn't have a license fee which makes their scheme cheaper for food companies. RA has been criticized for having too close relations with major buyers such as Kraft, not stepping in when Chiquita plantations violated the RA label standards and for being not strict enough, for example by allowing labelled coffee to have a minimum of 30% certified beans.
On the other hand RA claims that
In 2014, the State of Sustainability Initiatives Review completed a study which reviewed independent organizations on social, environmental, and economic indices. The Rainforest Alliance/Sustainable Agriculture Network scored the highest overall average (84 percent) in terms of social impact, above Fairtrade (73 percent), UTZ (58 percent), and Organic (51 percent). (source)
UTZ can be classified somewhere in between FI and RA (but closer to RA than FI). UTZ doesn't specify minimum prices. Instead they teach farmers sustainable and efficient farming methods which should increase yield and quality, and thus improve income. Farmers do get a premium for certified products, but how much has to be negotiated between farmers and the first buyer.
UTZ also teaches how to improve working and living conditions.
Use of pesticides is allowed by UTZ, if those pesticides are also allowed in countries like the US, European and Japan and not exceed the recommended quantities.
UTZ is the only organization of the 3 that has an independent third-party certification body for auditing and monitoring.
It's difficult to say which label is best. There are indications that FI is the most strict label and Rainforest Alliance the least, but all 3 labels have their strengths and weaknesses. Ultimately your label of choice will probably depend on which goal you feel is more important; social problems for farmers, maintaining biodiversity and ecosystems, or a combination of both.
An interesting development is that UTZ and RA announced a few months ago that they will soon merge and continue with 1 certification scheme under the name Rainforest Alliance. The new standard will be published early 2019.