First of all, we must note that there are tremendous numbers of differences between these two eras since the first lasted almost 400 years and the second lasted almost 700. Therefore, we can only look at differences in the broadest possible ways. Let us look at three major differences.
First, there was a difference in the nature of the government. In the Heian Period, government was very much a government of aristocrats. The government was officially headed by the emperors but was actually dominated by aristocrats from the Fujiwara clan. By contrast, during the Age of the Samurai, government was by the samurai class. They were military leaders and were not just from one clan.
Second, the Heian Period was one in which there was no influence from the West. The Heian culture was strongly influenced by China, but no Westerners had yet reached Japan. This was very different from the later period. In much of the later period, Japan and its leaders had to decide how to deal with western influences.
Finally, for much of the Age of the Samurai, Japan was united. By the end of the Heian Period, there was a great deal of power in the hands of local warlords. This meant Japan was not effectively united. This continued into the Age of the Samurai but then ended when Japan was unified by Oda Nobunaga, Toyotomi Hideyoshi, and Tokugawa Ieyasu.