There can be differences, but often they are exactly the same thing. It's just 'current sense' resistors are really low values. As far as I know there isn't any generally understood difference between a current sense resistor and any other low-value resistor of the same power rating.
You should probably stop reading here.
Sorry, but I just have to keep going...
That said, I can think of two features you might encounter which are targeted primarily at current sense applications:
1. Four terminals. For high precision, and to mitigate parasitic resistances, two terminals are used to carry the current, two others are used to sense the voltage. This is sometimes called a Kelvin connection. These are unusual, but available.
2. Low thermal-emf. Any junction of two dissimilar metals is a themocouple. Every thermocouple generates a voltage dependent on temperature. A surface mount resistor would have at least these, probably more, at each end: copper trace to solder; solder to nickel plating on resistor; nickel to resistive element. Since these are symmetrical at the two ends, you'd think that they would cancel. And they would, if everything were at the same temperature. But in a hot resistor there are inevitably thermal gradients, which cause the resistor to generate thermal EMF (voltage). In high current, low voltage applications, this can be significant. I found in my 13.5A driver, a 0.005 ohm resistor generated enough thermal EMF to alter the current setting by 0.2A or more. This was most disturbing when I commanded a change from 10A to 0, and it went to zero then over the next 20 seconds or so, rose back up to 200 mA, then gradually went back to zero.
Both of these are rarely required, and substantially expensive due to low production volumes, as well as more complex design.