What word of two mentioned above should I use when I mean a map route which begins and ends at the same point?
Looking at your question, I feel you want to mean that there's a map where you begin at one place and also end at the same.
Cycle does not fit and loop refers more to a shape than what you probably want. Okay...
One of the common words used for this is ...
circuit - A roughly circular line, route, or movement that starts and finishes at the same place
You may also use track but I don't think it starts and ends at the same point whereas circuit does probably because the electronic circuit happens only when it completes its path.
I don't know of any single English word that would be understood to mean what you are asking without further explanation.
The closest phrase I can think of is "round-trip route".
You could use "loop" to describe such a route. You'd have to specify that you're talking about a route -- a "loop" can be anything that goes in a circle or something resembling a circle. Like you can say, "I travelled in a loop" or "The route made a loop."
"Cycle" implies something that happens over and over. We talk about the "cycle of the seasons" or the "election cycle". If you said, "I travelled in a cycle" ... Well, first people would likely think you meant, "I travelled ON a cycle", meaning a bicycle or a motorcycle. But assuming you used another wording or emphasized the "in", they would understand you to mean that you travelled the same route over and over.
@Maulik's suggestion of "circuit" is possible. I think in general, though, people would understand a "circuit" to mean a route that has many stops. Like we say, "The salesman made a circuit of his five biggest clients", meaning that he visited each of the clients, then went back to the first and started over, probably many times. We used to have "circuit judges" in America, which were judges who would travel through a series of small towns, each of which was not big enough to need a full-time judge. He'd arrive in a town, hear whatever cases had come up since his last visit, then go on to the next. Some judges still have an official title of "circuit judge" but they no longer ride from town to town on a horse. They just have authority over a large area.