What are the advantages and disadvantages of incumbency?

To answer this question, it's important to understand through whose lens we are judging incumbency and at what level of office the incumbent currently holds. For the sake of this answer let's assume that the incumbent is someone high-profile (i.e. a US senator).
From the incumbent's perspective, the primary advantage is the incredible amount of free publicity that comes with being the incumbent. You are already a part of the political machine and, depending on your station within the senate, you have the ability to hit the CNN/MSNBC/etc. talking head circuit to discuss either your policy views and your intentions for office, current and future—or, cynically, whatever will get people to vote for you in the next election. A secondary advantage of incumbency is that if you've made friends as a senator you get, more or less, free support from other people who share your public reach.
Conversely, and perhaps ironically, the aforementioned advantage can simultaneously be the primary disadvantage of incumbency. Because of your visibility, everything you do and say is under a microscope. All your previous votes, snippets from speeches, appearances, etc. are potential fodder for opponents and a 24-hour news cycle that feasts on controversy. In addition, if you haven't made friends in office (or you just so happen to be a weak candidate), your fellow congress-people will not hesitate to fan the flames of discontent of your stint in office.
In conclusion, the advantages and disadvantages of incumbency are, while well-documented, functionally immaterial. Incumbents rarely lose (see reference) primarily due to the never-ending fundraising they do while in office (thus come out of the gate with a massive financial advantage over opponents).
https://www.opensecrets.org/overview/reelect.php

Historically, incumbents have several advantages in elections, although the extent of that advantage varies with the specifics of the election and its locality, with the advantages being greater in specific countries under specific circumstances.
In many countries, a major advantage is that incumbents can affect the timing of elections, calling them when their polling numbers are favorable, although in some cases, such as the recent election called by Theresa May in the UK, that strategy can be unsuccessful.
A universal advantage of incumbency is that it provides a platform for massive amounts of free publicity. While challengers must work hard to get media attention, an incumbent has wide coverage in the press. This also means, however, greater scrutiny, which can reveal negative elements in a candidate's background.
Next, an incumbent is a known quantity. This can be an advantage when voters feel that the incumbent is doing a good job and a disadvantage when voters are dissatisfied.

If we look at incumbency from the point of view of the electorate, an advantage to repeatedly reelecting the same official is that this individual will eventually gain more and more power through seniority. This power can be used to help incumbents in the home district.
On the other hand, incumbents can get stale and entrenched. They can become too aligned with their corporate donors and stop paying enough attention to the desires of their constituents, especially if they can take reelection for granted in a district that is heavily weighted in favor of their party. Incumbents can also get entrenched in one way of thinking about problems and not be sensitive to new ideas or to changes in the air.
Stability brought about by long incumbency can be useful because of the access to power it brings, but if elected officials become too out of touch with the electorate it is often time for a change and some fresh blood.

The chief advantage of incumbency is that of name recognition. An incumbent running for re-election no longer has to go through the long, arduous, expensive business of trying to establish their name in the public eye. During their time in office, incumbents have constant access to the media, giving them the opportunity to get their message out and to justify their positions on the political issues of the day. As elected officials they're also entitled to a substantial grant of tax dollars to help them communicate with their constituents.
It's often been said that money is the mother's milk of American politics. Being an incumbent gives you easier access to the funds necessary to carry out a re-election campaign. Potential donors no longer feel that they are taking a chance on a rookie candidate; now they're backing someone who's already been tried and tested in the bear pit of political life.
As to the disadvantages, well name recognition is a double-edged sword. Lots of people may know all about you, but familiarity breeds contempt, as they say. Once an elected official has gained a particularly negative public image, then it's nigh impossible for them to change perceptions. If there's a desire among the electorate for something new, then an incumbent can look decidedly old hat.
A negative track record is also a disadvantage. Your opponents can go through your policy statements and votes with a fine tooth-comb, highlighting anything they think can be used to undermine you.
Voter fatigue is another potential pitfall for an incumbent. If the economy's in trouble, or if there's a general sense that the country's going in the wrong direction, then incumbents are particularly vulnerable to a public backlash. They're part of the political establishment, and as such the voters might very well be keen to give them a bit of a kicking come election day. The word "new" is one of the oldest in politics, and when there's a sea change in the air, it's pretty hard for incumbents to do anything about it, despite their manifest advantages.