To make things clear
First of all i repeat saying that I'm no expert on this matter and my answer is based on my interpretation of this article in Arabic (in addition to other references I found). I try to transmit what I've learnt or understood as is, so you will see me pointing at critical points and at discussions, but I won't tell you whom might be right or wrong. For example: when it comes to the discussion of the relevant angle for fajr and 'isha timing I'll just transmit the information as precise as I can as different references use different definitions or terms, so I'm not even sure whether it's the same angle they are quoting.
Note as I said in my comments the views of the madhabs play a role in the difference of the calculation or prayer time. And I'm not sure if and to what extent Muslims in Muslim country use the discussed calculation methods, as they might have a good database due to observations, based on the knowledge of the time ranges or limits for the prayer times according a madhab see for example in al-Muwatta'.
Prayer timings for the daily prayers and necessary parameters for the calculation
- The easiest two prayer times or at least time "ranges or limits" which are important for us when it comes to prayer times (and those where the different calculation methods mostly agree easily) are the time for sunrise and sunset (maghrib) الشروق والغروب. Because the calculation formula is well known and is only based on the knowledge of the Longitude (See for example here, you may find the information i give here there also for almost all the timings).
Note again for maghrib the shia' view is totally different and would need a special formula, as they say: as long as the redness in the eastern sky appearing after sunset has not passed overhead, Maghrib prayer should not be performed. It is usually taken into consideration by assuming a twilight angle like Maghrib = Dhuhr + T(4). (Same source as above).
With the knowledge of these two times one can easily conclude the time of dhor/zuhr prayer as it is the middle of the day between sunrise and sunset.
Now when it comes to 'asr prayer where we have differences between the madhabs:
Hanbali, Shafi'i and Maliki schools do pray 'asr when the length of the shadow of a thing is as long as thing itself, while the Hanafi school prays it when the shadow is twice as long as the thing itself. And for both there is a specific and well known equation to calculate the time.
The most difficult prayers for the calculation are the times for 'Isha' and Fajr/Sobh prayer: The major problem here is that beside the Longitude the equation needs the Angle of the gradient (or deviation) of the sunbeams زاوية ميلان أشعة الشمس which not only differs from location to location but also from season to season and the more we move away from the equator this deviation changes more dramatically. An other not negligible factor for the calculation is the altitude of your location. But the main Issue with using applications of programs which calculate the prayer times is that this angle is (in almost all applications supposed as) fix, which is a totally wrong assumption, as it is a variable value, as explained:
For example the method of Um al-Qora: uses an angle of 18.5° for fajr prayer while the ISNA (Islamic Society of North America ) uses 15°. In a thread discussing the matter somebody reported a statement of a specialist pretending that a more correct value would be 14° or even better 13.5° unfortunately I couldn't find any explanation or real proof for this statement so far telling us on what basis this angle has been calculated nor where this value applies.
The problem with fajr and 'isha'
Note just to let me give a true-to-life example of the significance of such a difference: based on angle calculations of 15° or 18.5° for locations around the Arabic Peninsula, the time difference (error) is of up to 30 min. Which means at some location one may pray fajr or 'isha' 30 minutes earlier than the exact timing (with the wrong value of an angle).
Here a quote for the calculation for 'isha and fajr timing from moonsighting.com
Above 48.5° (e.g., Vancouver, Canada), the sun does not go 18° below horizon on the longest day of the year.
Above 51.5° (e.g., Cambridge, UK), the sun does not go 15° below horizon on the longest day of the year. On other days, Isha calculated at 15° will give Isha time 2.5 hours after Maghrib. This becomes hardship.
Above 54.5° (e.g., Copenhagen, Denmark), the sun does not go 12° below horizon on the longest day of the year. On other days, Isha calculated at 12° will give Isha time 3 hours after Maghrib. This is even more hardship, so it is impractical.
Also note that the calculation method of Um al-Qora was lately (2008) revised from 19° to 18.5° and before that a revision for the time for 'asr prayer has been made based on one single "measurement" by observation of one single (sent out) comity!
It is exactly this assumption of a fix angle that creates problems, as a fix value could fit for some region of the world better than for an other, while it could be totally wrong elsewhere: So the method of Um al-Qora might apply for the Arabic peninsula but not at all for Europe or North America. On the other hand even if we assume we found a value for that angle which fits for the summer, we might need an other for the winter and maybe even for each couple of months for the same region. Which might lead to errors of up to 90 min (before) the actual prayer time in northern Europe or America, if we fixed that angle according to Um al-Qora for example.
Hints for the right choice
So one should chose the right method depending on where one lives: So for example if you live in Saudi Arabia you could use any method even Um al-Qora as the possible errors could only lead you to pray after the actual time instead of before as stated in the example above!
For most countries of southern Europe and central Europe the calculation method of Muslim World League would fit with a reasonable level of accuracy which declines the further north one is living (Especially for the two critcial prayer times fajr and 'isha').
In Europe and America one can find some reasonable calculation methods -at least for a season or couple of months in the year- which must be adapted, as they apply only for at most one season. For example in Vancouver (Canada) most known calculation methods declare fajr/sobh prayer 90 minutes before its real time in May!
So the best solution is to get time tables or chart which are based on (local) observations of astronomers, which mostly (or hopefully) could be found in mosques. An other possible solution is the use of applications which change/vary this angle with the change of time and season. One of those methods is that one of moonsighting.com and one could easily download a prayer schedule for the whole year from the site (see how they calculate here). And there seem to be an other app called Sky Prayers which uses the same method.
Note That some calculation methods instead of using a formula for the calculation (the deviation angle) of 'isha and fajr prayer times add or subtract a fixed amount of time:
Um-al-Qora uses 'isha'= maghrib + 90 min during the year with the exception of 'isha' = maghrib + 120 min during Ramadan.
- This site only provides calculation formulae in the Arabic (version) but prayer times for a lot of cities world wide.
- A site in Arabic with bit more difficult and (apparently) precise calculation formulas for the prayer times (maybe only an attempt to make things better). Which seems to defend the Idea that there's no logical reason to reject an angle value of 18° (reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann, U.S. Naval Observatory, 1992.) and calculate based on it.
Finally a practical hint: In most local newspapers (at least in Germany) you may find each day a timing for sunrise and sunset if these times are approximately (+- max. 5 min) the same as to what you find on a your prayer chart, then your charts are useful, as I've read on the site of moonsighting.com that maghrib time (sunni) is (theoretical->should be newspaper value) sunset + 3min or (shia or ja'fari) sunset (theoretical) + 17 min.. So at least for 3 prayers (dhor, 'asr, maghrib) you could be sure that you have a useful timing.
I hope I could help to some extent.
UPDATE well I just read this article in Arabic which shows that -at least- in Germany fiqh scholars and astronomers are encouraged to work together on updated (corrected) prayer time tables for 2017!