Both Cuba and Mexico have had multiple revolutions. For the purposes of your question, I will compare the Cuban Revolution of 1898 and the Mexican Revolution of 1821, both of which were fought for independence from Spain.

The Mexican Revolution grew out of the ideas put forth during the Enlightenment advocating a people's right to self-rule based on a representative government. Starting with the Grito de Dolores 1810, a large movement led at first by Mexico's indigenous and mestizo populations began fighting for an end to Spanish rule. Their motives were to achieve better representation and to secure the rights of all Mexicans. They were eventually joined by wealthy criollo Mexicans, who joined the struggle for independence in order to maintain their positions of privilege once independence was achieved.

Throughout the late 19th century, Cubans were also fighting for independence from Spain. Like Mexico several decades earlier, they were fighting for home rule and better rights for the Cuban population. By this time Cuba and Puerto Rico were all that remained of the Spanish Empire in the Americas. In an attempt to maintain their power, the Spanish administrators of the island were becoming increasingly harsh and despotic in keeping order. This pushed Cuban intellectuals and land-owners to call for self-governance, if not outright independence. Like in Mexico, the Cuban Wars for Independence were fought by Cubans of all socioeconomic backgrounds and races. Also, like Mexico, the Cuban struggle for independence was mostly fought on racial and social grounds, but with the aid of the elite classes who saw the opportunity to vault themselves into positions of power once the dust settled.

Although there had only been a single Cuban Revolution, there have been several revolutions and revolts in Mexico. So for this question I’ll be comparing the Cuban Revolution to the 1821 Mexican revolution.

The Mexican Revolution broke out in 1821 and was fought between the Spanish and a ragtag alliance of Indians, Mexican-born Spaniards and Mestizos. It began as a peasant-led rebellion against the Empire of Spain and ended with an alliance between royalists and guerilla fighters. The revolution broke out in a small village known as Dolores where a priest, Miguel Hidalgo, helped lead the militia of his town against the Spanish. Hidalgo would eventually be capturing and killed. Guerilla fighting ensued for about a decade until an alliance was formed with powerful loyalists, who eventually helped the rebels push the Spanish out of Mexico.

The Cuban Revolution began in 1953 when Revolutionaries, most of whom were native Cubans, attacked a military barracks. Unlike the Mexican revolution, this was an internal struggle between a left-wing revolutionary group and a semi-totalitarian regime that was viewed as corrupt and the puppet of western interests. Like the Mexican revolution the instigators eventually failed in their first attempt, but after they were freed they were exiled to Mexico where they planned future attacks with an alliance of like-minded revolutionaries from other Spanish speaking areas. Most notably Che Guevera of Argentina. A prolonged guerilla war began, similar to that in Mexico. After repeatedly defeating the better armed military, a counter-offensive launcher by the rebels succeeded in forcing the Cuban president, Batista, to resign.   

How were the motives of the Cuban and Mexican Revolutions similar?

I assume that you are talking about Fidel Castro's revolution in the 1950s as opposed to the rebellion against Spain in the 1890s.  And I assume you are talking about the Mexican Revolution of the 1910s in which the government of Porfirio Diaz was overthrown.  The motives in these two revolutions were similar in that both sets of rebels wanted to overthrow a long-standing dictatorship that had (in many people's minds) been run for the benefit of foreigners and a small native elite.

In both cases, there was a dictator who had ruled for a long time--Porfirio Diaz, who had ruled for over 30 years in Mexico, and Fulgencio Batista, who had ruled Cuba for most of 26 years.  Both dictators were seen as people who ruled in ways that were of great benefit to foreigners, particularly business interests from the United States.  Both were seen as rulers who helped only a small sliver of their own population.

Because of this, each dictator faced eventual rebellion from people who wanted a government that would be more responsive to the broader population of their country.  This desire for a more responsive government motivated both the Mexican and Cuban Revolutions.