Sometimes it matters greatly, sometimes it matters not at all.
I've been on the receiving end of "it matters greatly" a couple of times.
- I was a technician in a Motorola factory that built the CQM6000. The command board (microprocessor and audio signal processing) could be placed in a test mode by attaching a pull-up to a specific accessory connector pin at power on. The text jigs all used a pull-up to the 12V power supply. This worked fine for thousands of radios - until one day it didn't. That digital signal went to the input of a 74XX something or other latch operating at 5V. The sudden change from "test mode works" to "WTF?" was that somebody, somewhere, bought that latch from a different supplier. The new ones didn't like that 12V pull-up on a 5V input. The old ones shouldn't have worked, either, but they were more tolerant. I was the lucky fellow who got to track down the cause of the test-mode failures. I knew how the test-mode worked because I had a test rig that made use of it. All the radios with test mode failures worked on my rig, but not in other ones. It turned out that mine used a pull up to 5V like it should have where other test rigs used a pull up to 12V. The pull up to 12V was used because the input pin and the 12V pin were right next to each other in the accessory connector. The test rig plug had a 1206 resistor soldered to the pins to activate the test mode. Once I figured out what was going on, all the test rigs were modified to use a proper pull up to 5V. Problem solved -but production was slowed for several hours while we looked for the cause and got it fixed.
- A small company I worked for had a custom controller board made for a repeater built of a couple of MC Micro (M110) mobile radios. The existing controller for the repeater used 9V digital logic, and the stuff we needed to connect to it used 5V digital logic. The board that the company had made by an independent contractor used some 4000 series CMOS ICs in the interface between the two systems to bridge the 9V/5V gap. The prototype worked, so we had the whole series manufactured. The finished boards didn't work. The ICs in the protoype and the ICs in the final product were from different manufacturers. The ones in the prototype had different voltage levels for high/low than the ones in the final product. We had to buy a bunch of ICs (from the correct manufacturer) and replace them in all of the finished boards because we didn't have time to have the controller redesigned.
Another case for "it matters greatly" is when selecting coupling capacitors for radio frequency circuits. A 100pF capacitor isn't just a 100pF capacitor - if the frequency is high enough. At high frequencies, you have to look at the inductance and self resonance of the capacitor. Two optically identical capacitors with the same value can be totally different at high frequencies.
A case for "it doesn't matter" are the typical pull-up resistors in digital circuits. Pretty much any 10k (or whatever) resistor will work. You don't need high precision or low inductance or any thing else special. Just (approximately) the correct resistance and the circuit is "happy."
You need to have some idea of what's critical and what's not.
Take the examples above.
The problem in the factory wasn't something that should have been anticipated. The service manuals all said "pull up to 5V for service mode." The guy who wired the test rigs saw the 12V handy (and 5V more difficult to arrange) and wired it the easy way. It worked, so nobody complained.
The repeater problem should have been designed to be more robust, though, so that it wouldn't matter which IC was used. It depended on the "typical" voltage levels for high and low rather than being designed with the maximum variations in mind (which the datasheets also give.) 5V was right at the edge of the low end of "high" for the ICs used, so the design should have considered it to be a critical point and either specified a specific IC - or better, taken a different route for converting the logic levels.
To get back to your question, take the example of the 555. If you are using it in a typical circuit, then it probably doesn't matter whether it's an LM555 from TI, a UA555 from Fairchild, or whatever. If you are going for very long timer periods or low power operation then you might need to specify a TLC555 (a CMOS variant of the 555 designed for low power consumption.)