One of the german languages "specialities" is that it uses a lot of compound words. This is true for nouns (ie. "Hand"="hand", "Tuch"=cloth", "Handtuch"="towel"), but it is also the case for verbs.
There is one difference between compound nouns and compound verbs, though: while compound nouns are most ly formed combining (two or more) simple nouns compound verbs are created combining simple verbs with prefixes. This is what you noticed that they are similarly written but the beginning is different. The prefix will modify the "basic" meaning of the simple word.
Here is an example:
laufen (to walk, to run, to go)
weglaufen (to run away)
ablaufen (to run its course)
umlaufen (to run around as in: to avoid comtact, but also to run over)
[sich] verlaufen (to be/get lost, to lose ones way)
verlaufen (to blend, but also: to take course, similar to "ablaufen")
and so on. Your example "absetzen/aufsetzen" is similar: the simple word is "setzen" which means "to sit" or "to place/to position". From there you get:
absetzen (to set down, to decant, also "jemanden absetzen" to remove so. from office)
aufsetzen (to put something, to superimpose, to don sth.)
versetzen (to transfer, to displace, but also: to pawn sth.)
hinsetzen (to sit down)
and so on. Here is a (also not even near exhaustive) list of prefixes verbs are combined with:
Also notice that sometimes the meaning is quite figuratively. For instance, there is "stehen" (stand) and there is "geradestehen" (basic meaning to stand straight or to stand upright), but the real meaning is either to take responsibility ("für etwas geradestehen" = "to take responsibility for sth") or take responsibility on someones behalf:
Ich werde für seine Schulden geradestehen.
means: I will pay his debts (if he defaults on them).
Addendum: compound verbs are either separable or inseparable (unseparable?). For instance, the last example I gave is a separable verb:
Ich werde geradestehen.
Ich stehe gerade.
Others are not:
Ich verstehe. ("stehen" + "ver")
Ich werde verstehen.
Ich habe verstanden.
Ich stehe auf. ("stehen" + "auf")
Ich werde aufstehen.
Ich bin aufgestanden.
There is (to my knowledge) no simple rule for what is separable and what is not. It depends on the prefix, i suppose, at least I can't think of a negative example. Compounds with "ver-", "be-" and others are always inseparable, "auf-", "mit-" and also others are separable.
Addendum secundum: as @RDBury pointed out in his comment, there are compound verbs which separate on certain occasions and don't on others. Examples would be "umlaufen" and "umfahren". It can mean "to run/drive over" or "to run/drive around [sth.]"
Ich umfahre die Person. (I drive around the person.)
Ich fahre die Person um. (I drive over the person.)