The cable for an oscilloscope probe (connected to the a high impedance oscilloscope input) is (generally) not a standard coax cable. If it were, say, a standard 50 \$\Omega\$ coax cable, then when a sharp-edged or high frequency signal reaches the high impedance input of an oscilloscope (typically 1 M\$\Omega\$), the signal will be reflected back down the cable. This can not only result in undesired artifacts in the oscilloscope's graphic output, but might also affect the behavior of the circuit under test.
[Note: Many oscilloscopes also have a 50 \$\Omega\$ input. A standard 50 \$\Omega\$ coax can be used with this input, but the circuit under test must be of the sort that supports such a low impedance load, (for example a LISN).]
To avoid the undesirable effects of impedance mismatch between an oscilloscope probe cable and the 1 M\$\Omega\$ input impedance of an oscilloscope, oscilloscope probe cables are (often) made with a high resistance central conductor. The resistance of this conductor is not so high that the overall impedance of the oscilloscope plus cable is substantially increased above the standard 1 M\$\Omega\$, but is high enough to dampen reflections in the cable.
An article by Doug Smith The Secret World of Oscilloscope Probes discusses the construction of oscilloscope probe cables.
But I also saw that sometimes a coaxial cable is connected between the DUT (device under test) and the oscilloscope. When is better to use a coaxial? Why would I use a coaxial instead of a probe?
Use a standard 50 \$\Omega\$ coaxial cable if your oscilloscope has a 50 \$\Omega\$ input connector and either the circuit under test is designed to support a 50 \$\Omega\$ load being attached to it or you have a special "low impedance" probe designed to be connected to the 50 \$\Omega\$ input of the oscilloscope, and your circuit supports probing with your "low impedance" probe. (Typically, the total impedance of such a probe plus the cable is 500 \$\Omega\$).
You might also be able to use a 50 \$\Omega\$ coaxial cable between a circuit and the 1 M\$\Omega\$ input of an oscilloscope if the frequency is low and the edges of signals are not too sharp. (Otherwise you might get ringing).
Use a standard probe when the circuit under test does not support having a low impedance connected to it, or the signal frequency is high, or signal edges are sharp.
You may use a dedicated coax connector as a test point in a circuit if you have a coaxial cable that mimics the behavior of a standard probe and cable. (I will not describe how to mimic a 10X probe). The cable in this case should NOT be a standard 50 \$\Omega\$ coax, but should have a resistive center conductor, as described above, and in Doug Smith's article. (I have no idea where to obtain such a cable).