Do people calling my cellphone hear a different ringing (a.k.a. ringback tone) when I am abroad?
Short answer: very likely. Countries often do not use the same sound frequency and cadence for the sounds we hear when dialing a phone number: ringback tone, busy tone, and so on.
The International Telecommunication Union publishes a PDF document describing ringing tones used by national networks across the world. To compare ringback tones between different countries look at the sound frequency and cadence they use for their various types of tones. For example, the US uses the familiar 2 second ring followed by a 4 second pause while, when dialing UK phone numbers you will hear a double ring followed by a 2 second pause.
To actually hear some of the most common ringback tones used across the world visit this page: https://en.wikipedia.org/wiki/Ringing_tone.
What if I don't want people calling me to know that I am abroad?
When running a business or for various other reasons you may not want people dialing your phone number to know that you are in another country. Using a VoIP smartphone app is one way to do this. For example, according to a relatively recent discussion on a UK forum you can do the following: sign up for a Skype number, redirect calls to your current mobile number to ring to the new Skype number and answer calls using the Skype app installed on your smartphone. You can either set up the redirection using your phone settings or by contacting your wireless service provider. Your wireless number and the new Skype number must be from the same country.
For the above to work you would of course need to have an Internet connection to be able to answer calls. Other VoIP services may work in the same way provided they are able to give you a phone number and have an app you can install on your phone.
If you'd rather not use an app, a second solution could be a Call Forwarding service which specifically advertises local ringback tones as a feature. Here's one such provider I was able to find (there may be others). In this case however, people from back home would have to dial your new VoIP number which would redirect calls to your existing mobile.
So I guess it all depends on what you want to do and whether you are looking for a temporary or a more permanent, business oriented solution. With any of the above services be sure to first test if it works as expected. Ringback tones can be tricky, there are countries where at least two different tones are used depending on the service provider.