internet music same for any tuner?

AFAIK, your tuner has nothing to do w it. Its not a radio signal at all, but a digital data stream. Internet "radio" is just a reference to a "broadcast" of music over the internet, available to anyone w a web browser and a DAC/audio card.

Streaming internet music has pretty much replaced my tuner for home listening. I still listen to the radio in my car. There are some really wonderful internet stations that don't have commercials. Radio Paradise being one of them.

Also most radio stations do stream their broadcasts over the internet which brings up another point: Some stations, like Radio Paradise stream at a much higher bit-rate than most local stations that often only stream at 48Khz. A decent DAC and high-end system will show you that this rate will not compete with the fidelity of a decent FM radio. At bit-rates 2.5 times (or more) this problem is solved.

A conventional am/fm tuner will not work for internet.

There are devices that provide the equivalent of an am/fm tuner as a high fidelity stereo system component for finding and playing internet radio stations as sources. These also generally allow for additional sources besides internet, an in house computer functioning as a music server, for example.

Roku Soundbridge (<$200, see my system here for an example) is one such device.

If you have a local college or NPR station (non-compressed), then a good tuner sounds as good or better than a Squeezebox/DAC. This is my opinion of course.
I use both a tuner and a squeezebox because I have a great jazz station in my city that I cannot pick up on the Squeezebox. The Squeezebox is well worth it for the vast variety of musical choices.

I grew up on music heard over FM radio.

But there is a much larger assortment of stations with very good sound in most any music genre on internet compared with FM these days, even in larger metro areas.

Plus, you do not have to worry about noise or other distortions due to weak signal, interference etc.

There is no comparison...internet blows away FM in most every regard already and the trend will only increase most likely over time.