A large part of the reason for my ability to rapidly pick up on Polish was the fact that it is a Slavic language, and as such, it shares a lot of similarities with Russian, a language which I already know quite well. I had already overcome many of the most difficult aspects of learning a Slavic language, such as noun declension and verb aspects. There are still many differences, and plenty to be learned — I myself hope to become fluent in Polish soon — but there's no denying that being a speaker of Russian gives a huge head start.
On paper, Russian and Polish don't look anything alike — they don't even use the same alphabet — so it would be easy to assume they're too different. But in actuality, the sounds in use are mostly the same. Russian is written in a Cyrillic alphabet, and there is only one sound per letter, with no digraphs, whereas Polish is full of digraphs and even has one letter (ż) with a matching digraph (rz) to make the exact same sound.
Most of the letters map up pretty easily:
Polish does away with the hard and soft signs, though it retains palatalization on the most common letters. This is accomplished by use of the grave over the consonant.
While Polish frees you from the burden of those softened Russian vowels, they do introduce three new ones: two nasals (ą and ę) and the ó, which is essentially just a reduced o. And there is also a new consonant ł, which is more or less a glide over the u, akin to the w in English, but completely missing in Russian.
In practice, a handful of generalizations make the switch from Russian to Polish pretty easy:
Russian ть becomes Polish ć
Russian ль becomes Polish l
Russian л becomes Polish ł
Russian ю becomes Polish ę
Russian р becomes Polish rz when followed by e
Working with this small handful of over-simplified assumptions, it's easy to quickly get
up to a working use of the language based on guesses from your Russian knowledge.
Сижу около реки.
Siedzę kolo rzeki.
Вижу автобус на улице
Widzę autobus na ulicy
There are really only a few challenges in going to Polish speech from Russian, and they're not difficult to get past.
The most significant difference is that, unlike Russian pronunciation, Polish does not have vowel reduction. Every vowel is pronounced in its pure form, as it is spelled. This feels a little weird coming from Russian, but it's easy — just say it as it's spelled. Few rules could be simpler.
Russian, like English is a stress-timed language. But Polish is a syllable-timed language, like Spanish. Every syllable gets more-or-less equal timing. This fact actually makes most things easier to pronounce, because it's actually wrong to shove a bunch of consonants together. Syllable-timing usually goes hand-in-hand with a lack of vowel reduction.
And, on the topic of stress, Polish words find their stress on the penultimate syllable. But unlike Spanish, this is almost completely, dogmatically, robotically, always true. So true, in fact, that there are no accent marks. I have yet to encounter a single Polish word which is pronounced with stress anywhere other than on the penultimate syllable.
The aspect that I find most difficult with Polish was the lack of soft vowels. Changing my ye to e proves to be quite a challenge, but it gets easier to remember after repeating a word correctly a few times.
Also, in conjuntion with that, the implicit softening of certain letters when followed by an i was difficult to remember. In fact, particularly in the case of the letters c and s, Polish softening seems to have a dual effect of converting them to cz and sz (respectively), and then softening them. I find it really easy to understand this when I hear it, but incredibly difficult to remember when I speak.
While there are a few constructs where the verb "to be" can be omitted in Polish, the copula is generally used, whereas it is almost never used in Russian, except for it's presence in "there are [thing]" constructs. As a native English speaker, this isn't terribly difficult to get used to, but I can imagine it would be as hard for a Russian as any other language is for them.
There are two grammatical differences between Polish and Russian which, in my opinion, give Russian a huge advantage in ease of use. The first of those advantages is the added attention to gender on plurals in Polish.
In Russian, genders are observed in the singular, but plurals are genderless. However in Polish, there is one form for plurals which include men, and another form for all other plurals. In practice the non-masculine plural is usually the same as the neuter single, so it's not difficult to form, but it's still difficult to remember the need to do so.
The second major grammatical difference, and no doubt the most frustrating, is in the formation of the past tense. In Russian, past tenses are easy to form, by just adding one of four endings to a verb stem to match the subject's gender. In Polish, however, you have 12 possible endings, consisting of six possible subjects plus two possible genders for each subject.
I have repeatedly found it quite difficult to remember this fact both on hearing others and in speaking myself... often to the point where even when I know a word, it doesn't register for me mentally. While everything else makes the transition a fairly easy one, this is the only detail which I have found to be a hindrance to understanding Polish as a fluent Russian speaker.
All in all, I think the path from Russian to Polish is a fairly smooth one — much more favorable than going in the opposite direction. I can imagine where a Polish speaker would have a lot of trouble going to the overly palatalized, stress-timed, and vowel-reduced Russian language with its unpredictable stress. Taking something that's unpredictable and making it formulaic is always easier than taking something formulaic and introducing a bunch of chaos.
Based on my experience of the four major Slavic languages, I find Russian and Polish to be the most distantly separated, with Czech and Ukrainian falling somewhere in between them. I imagine that with a strong knowledge of both Russian and Polish, a person should be able to drop into Ukraine, the Czech Republic, Slovakia, or Belarus and immediately be able to communicate with little or no trouble at all, and picking up Ukrainian, Belarussian, Czech, Slovak, Macedonian, Bulgarian, or Serbo-Croatian should be extremely easy tasks.