"pick it up" tells you about the learning process: casual acquisition of skill through going about your daily life or doing your daily work. "get the hang of it" focuses on the result; it does not tell you anything about the learning process.

To be absolutely clear, your sentence does not mean "you will understand the manual...". It means you will successfully learn how to use the computer as you go about your daily work interacting intuitively with the computer ( and/or asking your colleagues about it from time to time.)

Yeah there is. 'Pick it up' can have more than one meaning. For example, pick up that pen, please. Literally means, pick it up with your hand, but in this context (with the computer manual), it means that you will understand it. So it's more of an abstract meaning I guess. Hang it up is completely different. Hang up your coat (put it in the wardrobe, or on the coat rack), or sometimes it might be used to say that someone is quitting or retiring ...For example, Messi has decided to hang up his boots (retire from football).

Another important difference is the difficulty involved. If you 'pick up' something it suggests that it is relatively easy. If you 'get the hang of' something there is an implication of an initial difficulty - or perhaps only an initial reservation on your part - which you overcame at some point.

pick it up means becoming able to do something in the course of daily life or daily work. get the hang it means learning to do something (reasonably proficiently). The two are not the same but you could say that they both point to the same outcome, which is that you will have learned to do something.