I have an offer for a Masters in Mathematics from Technical University of Munich.. and I have also applied in Bonn and Berlin Mathematical School (this one is for Phd)..

Does anyone have experience of these universities/cities ? Which program is considered the best? (I do not have a preference for a specific program's content. )

Which other universities could I consider? Any thoughts or advice would be helpful..

I am a little surprised you applied for a Ph.D., as you cannot generally pursue a Ph.D. in Germany without obtaining a Masters-level university degree first.

Technical University of Munich is one of the best universities in the STEM fields in Germany and has officially been designated a "University of Excellence". However, take into consideration that Munich has the highest cost of living of any city in Germany, and finding affordable accommodation can be almost impossible, in particular for students.

Other good universities in the south are Stuttgart and Karlsruhe, as well as Tübingen and Heidelberg. RWTH Aachen and TU Clausthal also have good reputations in STEM. Some of the universities in the east (Leipzig, Dresden, Weimar) are also really good and generally offer much lower costs of living.

As a general remark: At least within Germany, there really is no big difference in "reputation" between universities. Degrees from all public "Universitäten" are considered more or less equivalent. Only the Universities of Applied Science ("Fachhochschule", "Hochschule", "Technische Hochschule") are generally considered second class. The perception may be a little different abroad, of course. As a general rule of thumb, larger universities in larger cities and older "historic" universities are considered better. However, in my experience the variation between professors has a much greater impact on the learning experience than the "reputation" of the university as a whole.

If you want to specialize into one particular research topic, however, certain research groups at certain universities may be much more famous and helpful for your career than others. Obviously, you would have to pick a topic first...

you might also consider this:

'The German research base is characterised by non-teaching institutions and looks complicated, but also relatively well-funded, when compared to the UK’s. Its structure also leads to underperformance in the global league tables, making German research appear less good than it really is. For example, if the Max Planck Society were included in the Shanghai Jiao Tong University league table, they could displace Cambridge as the top placed European university and knock Oxford out of the top ten. However, there appears to be some convergence between how research is undertaken in the UK and Germany.'

from here:
the comment underneath the article is well worth reading too as it explains some differences between the english and the german system.

Another remark that might be helpful for you: At least in the classical structure of German academia, a Masters degree and a Ph.D. are completely different. For a Masters (former "Diplom"), you essentially go to school and take courses and seminars, complemented by a relatively compact thesis period at the very end. As Ph.D. "student", you essentially work 100% and open-ended as independent researcher and teaching assistant, without receiving structured instruction or (sadly often) much meaningful supervision yourself. (Admittedly, this strict duality has been softened a little in recent years by the introduction of Anglo-American-style "graduate schools", but these are still a very small minority AFAIK.) As a matter of fact, Ph.D. "students" are not referred to as "Studenten" at all in German, but rather as "Doktoranden" and are rather considered to be university employees (and technically they usually are), as opposed to the "Studenten" up to Masters level.

Thanks a lot for your answers! I will take them into account..

(As regards Berlin Mathematical School, it offers a rather unique programme for a European
university, as the Phd lasts 5years and follows the US's policy for Phd programmes. )