Much of our expansion in the 1800's was conquest and settlement rather than colonization. Manifest Destiny and our westward movement systematically depopulated Native lands in favor of permanent white settlements. This was similar to earlier British policy of colonization in the 1700's, but different from the actions of empires in the later 1800s.
By the late 1800s and early 1900s we had become an empire, much the same as the European powers, where we had taken colonies like Guam, Puerto Rico and the Philippines for both their strategic location and their resources, but not as colonies where we would settle large numbers of white Americans. This too was similar to how the European powers acted at that time.
The major way in which American and European expansion were similar is that both were motivated similar things. Both Americans and Europeans had some mix of a desire to get new markets and more prestige. Both had some amount of desire to "civilize" the newly taken territories.
As far as differences go, American expansion was generally somewhat more benign, at least from our point of view. For example, the US promised (eventual) independence to the Philippines as soon as they were taken in the Spanish American War. In addition, Filipinos were given American citizenship.