One of the best answer I know of for this question comes from user zAy0LfpBZLC8mAC at Hacker News. This answer is so good I'm just going to quote it as-is.
TCP has head-of-queue blocking, as it guarantees complete and in-order
delivery, so when a packet gets lost in transit, it has to wait for a
retransmit of the missing packet, whereas UDP delivers packets to the
application as they arrive, including duplicates and without any
guarantee that a packet arrives at all or which order they arrive (it
really is essentially IP with port numbers and an (optional) payload
checksum added), but that is fine for telephony, for example, where it
usually simply doesn't matter when a few milliseconds of audio are
missing, but delay is very annoying, so you don't bother with
retransmits, you just drop any duplicates, sort reordered packets into
the right order for a few hundred milliseconds of jitter buffer, and
if packets don't show up in time or at all, they are simply skipped,
possible interpolated where supported by the codec.
Also, a major part of TCP is flow control, to make sure you get as
much througput as possible, but without overloading the network (which
is kinda redundant, as an overloaded network will drop your packets,
which means you'd have to do retransmits, which hurts throughput), UDP
doesn't have any of that - which makes sense for applications like
telephony, as telephony with a given codec needs a certain amount of
bandwidth, you can not "slow it down", and additional bandwidth also
doesn't make the call go faster.
In addition to realtime/low latency applications, UDP makes sense for
really small transactions, such as DNS lookups, simply because it
doesn't have the TCP connection establishment and teardown overhead,
both in terms of latency and in terms of bandwidth use. If your
request is smaller than a typical MTU and the repsonse probably is,
too, you can be done in one roundtrip, with no need to keep any state
at the server, and flow control als ordering and all that probably
isn't particularly useful for such uses either.
And then, you can use UDP to build your own TCP replacements, of
course, but it's probably not a good idea without some deep
understanding of network dynamics, modern TCP algorithms are pretty
sophisticated.
Also, I guess it should be mentioned that there is more than UDP and
TCP, such as SCTP and DCCP. The only problem currently is that the
(IPv4) internet is full of NAT gateways which make it impossible to
use protocols other than UDP and TCP in end-user applications.