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What is the difference between thickness and influence?
If you have four stones in an area and the opponent has one, you have more influence there than him.
If you have a group with two eyes and part of it faces an open area, you have thickness.
If you have thickness you can fight with advantage and no risk. If you have influence, you can fight with advantage but with some risk.
There is a continuum between the extremes, and in particular experienced players will see that a position that is solid but does yet have two obvious eyes can actually make them easily, and so that position can be treated as genuine thickness. More interesting is the frequent case where an experienced amateur player considers a position as thick but a
very experienced player (i.e. a pro) sees that it has weaknesses and so is still only influence. In the latter case a pro is liable to add a stone to make it genuinely thick, whereas an amateur is liable to attack too early.
It is probably best to forget the advice to use thickness to attack. It misleads. The hierarchy of principles is (1) make sure your thickness really is thick; (2) keep away from thickness. The latter proverb applies to both players, and so you either extend far away and make a lot of territory, or if the opponent gets too close you imitate a Venus fly-trap. In other words, whether you attack or not depends on what the opponent does. You do not set out with the idea of "Hey, I've got some thickness, what shall I attack?"
Perhaps the best way to learn about thickness is to make lots of it. Play honte and other moves that prophylactically cover your defects. It is possible to overdo this, but it's actually quite hard to overdo thickness, as it usually pays off eventually, even (or perhaps especially?) in the endgame. Even if your opponent makes an apparently big moyo while you do this, you'll find you can do things like invade him with abandon, because even if your invading group has to run away, it can do no collateral damage to your already thick position elsewhere (and of course your thickness also constitutes a safe haven for the running group, and a reef for an over-enthusiastic hunter).
If you do overdo the thickness building, this will translate into overconcentration, which is quite easy to spot by yourself, without a teacher.
How do you use influence? This is still at a formative stage of the game. You use it to try to determine how the game will proceed. It gives you the initiative in that area. You can be the one to decide whether influence turns into e.g. thickness, a moyo, post-facto reducing or spoiling moves (i.e. at low cost to you, stopping your opponent from getting control of an area).
If, out of these various strategies, you choose to try to turn influence into thickness, learning how to do that is one of the most interesting learning trajectories in go. It is a stage a little before the honte and other defect-covering moves, and includes principles such as ijime (bullying) and forcing moves, but above all it means just going for it - it's like learning to swim when you are still too nervous to take your foot off the bottom. The go equivalent of combining water wings and taking your foot off the bottom is to experiment with some outward-facing josekis. But do remember thickness is ubiquitous, and is about so much more than josekis. It includes the middle game, e.g. Shuei's use of centre-facing L shapes, and the endgame. Playing I-Spy Thickness is another delicious game you can play while playing go.