Actually, there was a greater relationship between the two than the above answer indicates. Both movements originated with the reform movements of the Second Great Awakening; and the Women's movement actually resulted to a large extent from the exclusion of women from the Abolitionist movement.
Women were quite active in the abolitionist movement in the north and south, including the Grimke sisters of South Carolina. The Grimkes were told they should not speak on the subject of abolition, as it was un-ladylike. There was heated debate over the issue, and the American Anti-Slavery Society split as a result. Elizabeth Cady Staton and Lucretia Mott, the co-founders of the women's rights movement were also ardent supporters of the abolitionist movement. Both women travelled to London at their own expense to attend an abolitionist conference but once there were denied admission because of their gender. The reasoning of the conference's sponsors was that the subject was too delicate for the ears of ladies. This event led them to call the Seneca Falls Convention which first espoused the equality (and equal rights) of women. William Lloyd Garrison, the famous author of The Liberator actually left the American Anti-Slavery Society (of which he was a co-founder) because its leaders refused to espouse the cause of women's rights.
Thus, it is more nearly correct to say that the women's movement and abolitionist movement were indelibly tied to one another; one having predated the other by only a short period of time, and both born of the common movement of the Second Great Awakening.
This is a great question. Both movements grew in earnest during the same period of time. So, on the one hand, there are many similarities. For example, both issues were a part of the Second Great Awakening and the social gospel movement.
During this time, there were women leaders, schools for women in higher education, and a strong push for the abolition of slavery. I would say what also makes them similar is their under-girding theology. Calvinism basically stated that people were totally depraved and incapable of changing themselves. Only God can do this by his grace. The new theology that took root was called Arminianism. This stated that people have free choice and wills. This meant that they could help themselves. More specifically, this meant that a city or nation could be reformed. This shift in theological thinking lent support to all sorts of social activities.
In light of this, there were huge overlaps between the women's rights movement and the abolitionist movement. That tie, in my opinion, was Christianity. With that said, people are inconsistent. So, of course, a person could support one and not the other.