The Sidekicks material was published in the Unearthed Arcana column in December 2018, as playtest content.
The Essentials Kit, released as a Target exclusive in June 2019 and everywhere in September 2019, includes an appendix in its Rulebook that is about Sidekicks.
So, what changes between the UA version and the Essentials Kit version?
Best Answer
Various topics between the two versions vary.
Guidance and Prerequisites
Unearthed Arcana
For a player character to gain a sidekick, there are some prerequisites to be met. The sidekick can be any creature from the Monster Manual or other D&D book, and these prerequisites are necessary:
There is no mention of minimal or maximum party-size (aka number of players) that limits the number of sidekicks in a group of PCs.
Both the Expert and Spellcaster classes requires the creature to "have at least one language in its stat block that it can speak".
Essentials Kit
In the adventure booklet included in the Essentials Kit – Dragon of Icespire Peak – the Running the Adventure chapter recommends having a sidekick if you, as the DM, are running the adventure for a single player. The sidekick must be one chosen from the Sidekick Cards included in the box.
The characters presented in the Sidekick Cards are humanoids of various races with at least one level of one of the Sidekick classes. There are no other requirements in this version.
Leveling Up
Unearthed Arcana
A sidekick gains a level whenever the associated PC gains a level.
Essentials Kit
The Sidekick explicitly counts as a PC for the purpose of sharing experience points between the party members. The Sidekick rules says:
Hit Points
Unearthed Arcana
Whenever a Sidekick gains a level, it gains a Hit Die based on its creature size (d4 for Tiny, d6 for Small, d8 for Medium, etc).
Essentials Kit
The creature Hit Die is always a d8, as the Appendix deals only with the Sidekicks presented in the Sidekick Cards in the box, which are all humanoids.
Ability Score Improvement
Unearthed Arcana
There is no guidance regarding which ability score you should increase when a sidekick gains such feature.
Essentials Kit
For each of the three classes, there is guidance to what ability score should be increased. For Warrior is +2 to Strength, for Expert is +2 to Dexterity, for Spellcaster is +2 to Intelligence or Wisdom, depending on its spellcasting ability.
Sidekick Classes
The classes available to sidekicks are the same for both UA and Essentials: Warrior, Expert, and Spellcaster. They, however, have different features for each level.
Unearthed Arcana
The UA classes have a complete table of features, from levels 1 to 20.
Essentials Kit
The Essentials Kit only presents a table of features per level ranging from levels 1 to 6. This limitation is due to the adventure included in the box: Dragon of Icespire Peak is meant to be played from levels 1 to 6.
Warrior
This section compares the levels 1-6 of the Warrior class.
\begin{array}{|c|c|l|l|} \hline \textbf{Level} & \textbf{Prof. Bonus} & \textbf{Unearthed Arcana} & \textbf{Essentials Kit} \\ \hline \text{1st} & \text{+2} & \text{Second Wind, Bonus Proficiencies} & \text{Martial Role, Selected Proficiencies*} \\ \hline \text{2nd} & \text{+2} & \text{Danger Sense} & \text{Second Wind} \\ \hline \text{3rd} & \text{+2} & \text{Improved Critical} & \text{Improved Critical} \\ \hline \text{4th} & \text{+2} & \text{Ability Score Improvement} & \text{Ability Score Improvement} \\ \hline \text{5th} & \text{+3} & \text{Extra Attack} & \text{–} \\ \hline \text{6th} & \text{+3} & \text{Ability Score Improvement} & \text{Extra Attack} \\ \hline \end{array}
*By "Selected Proficiencies", I mean a preselected set of proficiencies. The Warrior from the Essentials Kit has the following proficiencies already chosen:
Both versions are proficient with all armor, shields, and simple and martial weapons.
Expert
This section compares the levels 1-6 of the Expert class.
\begin{array}{|c|c|l|l|} \hline \textbf{Level} & \textbf{Prof. Bonus} & \textbf{Unearthed Arcana} & \textbf{Essentials Kit} \\ \hline \text{1st} & \text{+2} & \text{Expertise, Helpful, Bonus Proficiencies} & \text{Helpful, Tools, Selected Proficiencies*} \\ \hline \text{2nd} & \text{+2} & \text{Cunning Action} & \text{Cunning Action} \\ \hline \text{3rd} & \text{+2} & \text{Jack of Many Trades} & \text{Expertise} \\ \hline \text{4th} & \text{+2} & \text{Ability Score Improvement} & \text{Ability Score Improvement} \\ \hline \text{5th} & \text{+3} & \text{Extra Attack} & \text{–} \\ \hline \text{6th} & \text{+3} & \text{Expertise} & \text{Extra Attack} \\ \hline \end{array}
*By "Selected Proficiencies", I mean a preselected set of proficiencies. The Expert from the Essentials Kit has the following proficiencies already chosen:
Both versions have proficiency with light armor and all simple weapons. The Unearthed Arcana version also gains proficiency with two tools of your choice, while the Essentials Kit version gains proficiency with rapiers, shortswords, and possibly with thieves' tools and a musical instrument (these last two from the "Tools" trait in its stat block).
Spellcaster
This section compares the levels 1-6 of the Spellcaster class.
\begin{array}{|c|c|l|l|} \hline \textbf{Level} & \textbf{Prof. Bonus} & \textbf{Unearthed Arcana} & \textbf{Essentials Kit} \\ \hline \text{1st} & \text{+2} & \text{Spellcasting, Bonus Proficiencies} & \text{Magical Role*, Spellcasting**, Selected Proficiencies***} \\ \hline \text{2nd} & \text{+2} & \text{Magical Recovery} & \text{–} \\ \hline \text{3rd} & \text{+2} & \text{–} & \text{–} \\ \hline \text{4th} & \text{+2} & \text{Ability Score Improvement} & \text{Ability Score Improvement} \\ \hline \text{5th} & \text{+3} & \text{–} & \text{–} \\ \hline \text{6th} & \text{+3} & \text{Potent Cantrips} & \text{Potent Cantrips} \\ \hline \end{array}
*The Essentials Kit Spellcaster has the Magical Role feature, which defines the Spellcasting Ability for the sidekick character. It only allows for Intelligence or Wisdom to be chosen.
In the Unearthed Arcana version, the Spellcasting Ability is chosen in the Spellcasting feature, with Charisma being a possible Spellcasting Ability too.
**Each version of the Spellcasting feature has a different spell slot progression (shown below), along with other differences:
\begin{array}{|c|c|c|c|c|c|} \hline \textbf{Level} & \textbf{Cantrips Known} & \textbf{Spells Known} & \textbf{1st} & \textbf{2nd} & \textbf{3rd} \\ \hline \text{1st} & \text{3} & \text{2} & \text{2} & \text{–} & \text{–} \\ \hline \text{2nd} & \text{3} & \text{3} & \text{3} & \text{–} & \text{–} \\ \hline \text{3rd} & \text{3} & \text{4} & \text{4} & \text{2} & \text{–} \\ \hline \text{4th} & \text{4} & \text{5} & \text{4} & \text{3} & \text{–} \\ \hline \text{5th} & \text{4} & \text{6} & \text{4} & \text{3} & \text{2} \\ \hline \text{6th} & \text{4} & \text{7} & \text{4} & \text{3} & \text{3} \\ \hline \end{array}
\begin{array}{|c|c|c|c|c|} \hline \textbf{Level} & \textbf{Cantrips Known} & \textbf{Spells Known} & \textbf{1st} & \textbf{2nd} \\ \hline \text{1st} & \text{2} & \text{1} & \text{2} & \text{–} \\ \hline \text{2nd} & \text{2} & \text{2} & \text{2} & \text{–} \\ \hline \text{3rd} & \text{2} & \text{3} & \text{3} & \text{–} \\ \hline \text{4th} & \text{3} & \text{3} & \text{3} & \text{–} \\ \hline \text{5th} & \text{3} & \text{4} & \text{4} & \text{2} \\ \hline \text{6th} & \text{3} & \text{4} & \text{4} & \text{2} \\ \hline \end{array}
***By "Selected Proficiencies", I mean a preselected set of proficiencies. The Spellcaster from the Essentials Kit has the following proficiencies already chosen:
Both versions are proficient with light armor and simple weapons.
Other details
Finally, it is important to note that the Unearthed Arcana content is playtest material, and therefore, can be unbalanced. The Essentials Kit version is official, but the lack of progress after level 6 limits the official rules for Sidekicks.