As a native Russian speaker married to a native of Georgia, I can say that the most common grammatical mistake is the ending differences on adjectives between masculine/feminine/neuter types. There is no such distinction in Georgian.
For example, "დიდი კატა" ("big cat") in Russian would be "больш**ая** кошка" or "больш**ой** кот", depending on whether you are talking about a female or a male cat respectively, while "დიდი სახლი" ("big house") in Russian would be "больш**ой** дом" - notice the difference in endings. This one of those things that people learn eventually, but it takes time - and the feel of the language.
At the same time, I'd say that the most common pronunciation mistake is the difference between hard/soft letter "l" in different words. For example, in Russian, word "слушай" ("listen") is pronounced as "slushay" - with hard "l", whereas you could more than often hear from Georgians "слюшай" ("sljushay") - with soft "l". And the same happens the other way around, for example, with word "соль" ("salt" - with soft "l" in Russian), which I hear Georgians pronounce as "cол" - with hard "l". I'm not too sure of the exact reasons, possibly because there is only letter "უ" ("u") in Georgian, which, when follows a consonant, makes that consonant hard-sounding. To make it soft-sounding, it would go with "ი", e.g. "იუ" after a consonant.
There's an old Russian joke about it (any Georgians reading this, please do not get upset, I have very deep respect for you and your culture):
Учитель русского языка в грузинской школе говорит: "запомните дети, поскольку понять это невозможно: слова 'сол' и 'мол' пишутся с мягким знаком, а слова 'пошель' и 'нашель' - без.
Roughly translated into English it would be:
Russian language teacher in Georgian school is saying, "Memorise this, children, because it is impossible to understand: words sol and mol (English salt and moth respectively, in Russian with soft 'l' at the end, here expressly with hard 'l') have soft 'l' at the end, while words poshel' and nashel' (English went and found respectively, in Russian with hard 'l' at the end, here expressly with soft 'l') have a hard 'l'.