(Note that in all European countries I know, the payments you are referring to are not really taxes – flowing to the state budget – but tax-like contributions to mandatory insurance systems, which is why I will mostly use the word “contributions” to refer to them.)
Arguably, the notion that employees “really” pay those taxes and contributions does not fully make sense. You might just as well consider that employers have to pay whatever wage result in a take-home pay that allows them to recruit workers that are good enough for their purpose and therefore that they are the ones “really” paying all taxes (together with the part of the wage that employees get as take-home pay). Who says that if these taxes and mandatory contributions were lower you would get all the difference as extra cash on hand and employers would continue to pay the exact same labour costs? In principle, employers might also pocket the difference, without changing their employees' take-home pay.
Whatever the case may be, it's true that both employers' and employees' mandatory contributions should in principle have the same effect, economically speaking. Certainly from a market perspective, employers would be expected to hire employees only if the marginal cost of one employee (that's take-home pay, “visible” and “invisible” income tax and mandatory contributions and all other costs) are lower than the increase in production they need to turn a profit from the product of that employee's work. If labour costs go up, it does not matter whether it's in the form of increased salary or additional employer contributions, it will be less profitable to hire someone and might price some workers and/or businesses out of the market.
Conversely and with the caveat that payments to mandatory insurance systems are really deferred income (and therefore that reduced benefits might need to be compensated through additional pay), employees who are ready to work for a certain take-home pay should still be ready to do it for the exact same price if all the “invisible taxes” would disappear overnight.
The reason why the distinction does matter is that wages (very much including the minimal wage) are defined before taxes but after employers' mandatory contributions in work contracts, collective bargaining agreements and the law. This means that when a government decides to raise the rate of the employers' contributions, they will increase labour costs, risk increased unemployment or the wrath of employers but not the discontent of workers who won't see any change to their take-home pay. Conversely, when it's the employees' contributions that increase, people will see their take-home pay decrease (or more often stall), while employers won't see any changes, at least in the short-term (i.e. before people switch jobs and negotiate other contracts, giving everybody an opportunity to adjust the wages they are willing to pay or receive to the new economic conditions). And if you do both at the same time, you can always go on TV to argue that “everybody has to pay their fair share” and hope for some psychological effect.