In developing countries, a mass protest could reasonably end up in the local politicians finding themselves in jail or worse, such as what happened in 2011 to Qaddafi. On the other hand, the governments of developed countries are only responsible to voters during elections. Therefore a protest by a small percentage of the population (even if large in absolute numbers) should be of little concern to those in power.

However we frequently see examples of politicians creating concessions under the pressure of protests - for example the French president recently agreed to change the scope of the tax increases under pressure of massive protests in Paris. What's the reason behind this? Why care about protests at all?

Stability.

Security is an illusion.

Take this to mean even security in the belief of a set of rules as protecting you or anyone else. If rules by themselves are enough then making murder illegal should solve all murder.

Protests mark civil instability. Any instability can be a potential weakness to be exploited. The instability of a protest can have far reaching effects.

Physical Security:
They matter because they are frequently the last line of peaceful action, if they themselves are even peaceful.

A protest left unchecked quickly becomes a riot. And riots are not good for politicians. Many people judge politicians by how well they handled protests as well.
And there are countless examples of protests involving the loss of life.

Political Security:
Rules only matter when enough people follow them. If a protest has enough support, it can lead to a coup. Even without the protest itself being a physical threat, they are a clear political message.

A common trait among protests is that they are usually very one-sided. Support for the protest is high while opposition for a protest is usually low. A small group of people feel passionately about something that the rest of the people don't feel strongly for or against.

This means that it is typically inexpensive politically to listen to a protest. Concessions can be made with protesters often without upsetting your own supporters too much.

History:
Countless protests throughout history have shown to have mattered. They are proven to be meaningful. Protests have also established themselves to be dangerous to the situation that produced the protest. In this sense, protests can be viewed as a symptom of an underlying problem that doesn't go away when the crowds disperse. Politicians know this.

Disclosure: this is my personal common sense speculation, not backed up by scientifically conducted poll of developed countries' government figures.

TL;DR:

• Protests makes the government less popular, lowering chances to be re-elected (or, where political forms allow it, things like no-confidence votes, coalition fracturing or other democratic ways of losing power)




• Protests have practical material downside consequences to the economy (even if the leader doesn't care for civic virtue reasons, they care because again re-election chances impact).




• Protests have risk of bad optics/events for the government




• Protests have chance to increase popularity of protester's views; the longer they go the more of it.

Details

1. 
   

   Protestors actually have real, meaningful impact. Both direct, and indirect.

   
   
   

   At the very least, economic (you shut down a big street in a big city, that negatively impacts a TON of economic activity by everyone around - both activity by protestors, by businesses affected, by people who can't commute, by tourists - with ripple effects from those issues).

   
   
   

   This has negative effect on the mood of people in the country directly (who would be unhappy from direct or indirect effects) as well as on people in the country indirectly (less economic activity means less income for them in general AND less taxes+more spending leaving less money to spend in the budget of city/country). Hopefully (from the view of government) affected people will blame the protesters, whose fault it is. In reality, people very often - validly or not - directly blame the government.

   
   
   

   At worst, protests can turn actually violent (specifically, as your question is probably regarding recent French protests - France in the past has a sustained history of major protests turning violent, with burning cars and attacks on police etc...).

   
   


2. 
   

   Also, generally people's happiness overall tends to correlate with how likely the current government in a democracy gets re-elected.

   
   
   

   So, any general unhappiness (discussed in #1 above) lowers the chances of re-election on average, even barring direct effects and direct unhappiness/blame from them.

   
   


3. 
   

   Professional activists have good ways of propagandizing their views such that they look sympathetic. That's what their job basically is.

   
   
   

   Allowing protests to go on makes the government look unsympathetic as a consequence, regardless of underlying truth/nuances/validity of protesters claims.

   
   


4. 
   

   Protests have great propaganda value for activists protesting:

   
   
   
   
   

   • Media coverage. MUCH media coverage. That's one of the main goals of most modern protests.

   
   

   • Morale boost for their side from "being active"

   
   

   • Other people being familiarized with their views/ideas (due to above media coverage).

   
   

   • They look "principled" if protests persist.

   
   

   • Additional benefit for protesters: this one depends on how one protests; but I personally generally tend to have a positive view of someone who cuffs themselves to a government building as a form of protest, even if not sharing their views. It shows conviction and willingness to sacrifice personally (as opposed to sacrificing other people's time/money/convenience). Other people are even less critical/discerning and start sympathizing regardless of the form of protest.

   
   
   
   



5. Protests may turn truly violent, and government will risk having a major incident with someone dying/injured as a consequence. Typically, government gets blamed for that.



6. 
   

   This one is more nuanced: sometimes, the government may actually agree with protestors but lack political ability to act on those views due to divided domestic politics.

   
   
   

   In that case, protests give them cover to do what they want to do anyway, either by actually shifting domestic political opinion (see bullets above) or merely by offering a CYA excuse.

   
   

In many developed countries, the top political leaders can be removed or sidelined by second-rank political leaders at any time, not just during elections. This may be a vote of no confidence in a chancellor, or a president being unable to find a parliamentary majority for his budget any more.

So if those members of parliament or party officials get a feeling that the top leadership isn't delivering success stories any more, they may decide to look for a new top leadership.

the governments of developed countries are only responsible to voters during elections

I think this assumption is simply incorrect.

Votes for "the government" (whatever that means concretely) may be the most regular and direct way of citizens to influence the actions of a government. However, there is a host of other ways how political decisions and careers can be influenced, albeit with a certain delay, between elections, for instance:

• Elections for other political boards than "the government" in question, for instance on the state/province level. For a given party, losing the majority in a state can well be interpreted as a general failure of the party's course, which can in turn weaken the position of the party on the national level.


• Members of "the government" are often covered by some kind of legal immunity, but that can be removed if both the evidence against some wrongdoings of a politician and public demand to act upon it piles up.


• Even without anything legally problematic, unhappy citizens may be more willing to spend the effort of digging up ... unfortunate facts. Such as inappropriate former statements uttered by people in certain positions. This can damage the credibility of the person's party as a whole, and at least stall the implementation of some plans if, for instance, opposition parties demand the removal of said person from their office in return for supporting the plan.


• As mentioned in o.m.'s answer, there are formal processes for removing politicians from office outside of the regular election cycle.


• Lastly, while it is said that a voter's memory has a short half-life, a general and repeated unhappiness among citizens might still add up to undesirable effects by the time the next election comes around.