In linguistics, the concept of a more or less modern language usually refers to its grammatical structures, not so much to the lexical field (word formation). Languages with many cases such as Finnish are regarded as more archaic. The more streamlined and simple -- the more modern.
I'm a linguist and this just isn't true.
Modern and archaic are not concepts used at all in any modern linguistics I know of for a very good reason - there's no correspondence between degree of morphological complexity and anything else.
The specific differences between Polish and Russian could well have something to do with political history.
Basically, Polish has been a mono-ethnic language for a long period of time while Russian is polyethnic.
This means that Poland has largely been restricted for conducting the internal business of a single ethnic group and has been free to create and/or maintain quirky features that outsiders find .... strange.
Russian, on the other hand, has a long history as a lingua franca between different ethnic groups (a process that did not stop with the collapse of the USSR - Latvians, Georgians and Kazakhs are still more likely to communicate with each other in Russian than any other language). This often means that quirky local features are lost in the interest of easier mutual comprehension.
A couple of differences between Polish and Russian might be related to this.
- Russian has less free word order (compared with Polish that is)
- Russian requires the use of subject personal pronouns even when the form of the verb makes the subject clear ( ja govoriu po russkij not *govoriu po russkij )
These two features seem only to have strengthened in recent history and are the kinds of changes that are typical in languages that are used across ethnic boundaries.
Some of the other differences (lack of the copula in the present tense, a number of 'verbless' setence types)_might_ be due to this is as well, but I don't know the historical record well enough.