Resistor Tolerance - are measured resistor with the same value but different tolerances the same?
If you're measuring the resistance in similar conditions to what it will experience in the circuit, then yes, you can use either resistor. If the circuit is going to heat up the resistor by 70C, then maybe not. Resistors, capacitors, and inductors can all be made out of different materials, each of which has its own properties. Here are some things that might matter aside from the component value and tolerance:


Power handling
Parasitic resistance, inductance, and/or capacitance (important at high frequencies)
Temperature coefficient
Max operating temperature
Max soldering temperature
Operating lifetime (very important for electrolytic capacitors)
Breakdown voltage and polarization (for capacitors)
Matching of multiple components (sometimes more important than tolerance)
Electromagnetic interference and compatibility (radiation/sensitivity)
Noise generation
Package size and weight (often related to power handling and parasitics)
Reliability of the supply chain


Usually only a couple of these will be major concerns in any particular application. For example, in audio systems, you're very concerned about noise. In automotive and aerospace, a high max operating temperature is required. In high-frequency design, parasitic inductance and capacitance are a big deal. For low frequency, low power, low precision circuits, just about any component will do.