Is there any difference between 学習する and 習う?
These are similar words with subtle differences.
学習 has a bit more formal sound than 習う, and the difference in their meanings derive from that.
For example, 学習 tends to refer to "at desk" formal studies you do at schools and institutions. Mathmatics, science, English, that sort of things. In contrast, 習う often refers to lessons and extra-school activities such as karate, piano, cooking, and so on (see Worthy7's answer for sentences.)
The border line is blurry, though.
For example, take Worthy7's sentence "向こうの大学じゃ、習ってなかったし" as an example. This refers to a study in an university. But in this case 習う is used because the form of speech is informal.
Likewise, 学習 can be used even if it doesn't involve studying at formal institutions, when a rigid sounding noun form is needed. For example, 生涯学習センター (life-long learning center) is normally just a community center that has short programs, but you don't want to call it 生涯習い事センター because it lacks dignitiy.
学習する appears to be "It is learning" in the following examples:
In both cases it is someone talking about something else learning and acquiring knowledge.
And 習う seems to be a more normal learning:
In these 3 cases the speaker is talking about themselves learning something.