As a native Ukrainian and a teacher of English (for 20 years already), I can give evidence of what is complicated in English from the point of view of Ukrainian or Russian learners.
In Ukraine, everybody speaks Ukrainian and Russian, so the idea of being able to speak different languages is present since the early childhood. However, Ukrainian and Russian are very similar, they don't have any principal differences, although differences are present on all the levels of the language, beginning with alphabet and phonemes and ending with syntax. When Ukrainians start learning English they are usually shocked with the idea of how enormously different a language can be from what they thought a language should be.
In Ukraine, the majority of English textbooks and dictionaries are in British English. The first difficulty a learner faces is the amazingly complicated sound system of English, especially the vowel system. Ukrainian has 6 vowel phonemes, /a, ɛ, i, ɪ, ɔ, u/, English (RP) has 20 (excluding triphthongs), which is about 3.5 times more than in Ukrainian. And they can be short and long!.. And there are diphthongs!.. All of them are different from the Ukrainian vowels, none is pronounced the same way, and they all should be distinguished in pronunciation!.. Half of the consonants are alveolar (no alveolar consonants at all in the Slavic languages). And the consonants are not palatalized before /iː/… Really? How can that be? And the aspirated stops… None of the Slavic languages has aspirated consonants, few learners ever master them. And those /θ/ and /ð/… Then goes the spelling… The Ukrainian spelling is pretty phonetic, but strictly prescriptive, especially punctuation, at school Ukrainians have to learn lots of long and complicated rules, e. g. when to write the negative particle together with, and when separately from the following word or where to put a comma and where a semicolon. In English you've just got to learn the spelling of each individual word and no obvious rules (actually, there is one spelling rule), which is a relief for some, but a baffling difficulty for others.
The English noun doesn't have any genders, a luxury which only learners of French and German can fully appreciate. And no case forms! Still, it has something that is so complicated that most Ukrainian and Russian learners of English never master it fully. As for me, I cannot dare swear I've mastered it as much as I would like to. That thing is the number-countability-definiteness complex. The very idea of the noun category of definiteness / indefiniteness is alien to the majority of the Slavic languages except for the closely related Macedonian and Bulgarian languages. In order to begin using articles correctly Ukrainians have to learn to analyze the reality in quite a new way, to change their way of thinking from the moment they learn about the existence of articles and to the end of their lives. Few succeed. I think this category is much more complicated than gender and case taken together, since remembering the gender and case forms of a noun is mechanical work, you use this or that form automatically, the syntax suggests which form should be used. Definiteness is different. It's up to you which one you choose, a or the, but! the meaning of the whole utterance depends on this choice of yours, and you cannot skip the choice since it is mandatory in English that definiteness / indefiniteness should be explicitly marked on every noun, which means you really have to learn the analysis of the reality for definiteness, a skill which is needed only when you talk or write in English, that is, which is useless for most learners who usually use English for reading or watching films and who talk English only at English lessons.
The English adjective is a real candy with just one single form (in the positive degree) as compared to 24 gender-number-case forms in Ukrainian and Russian and 48 gender-number-case-declension forms in German. But the English verb… Everything about the English verb is very complicated. Well, the Ukrainian and Russian verbs are also complicated, but they are complicated in an absolutely different way, their synthetic morphology is tangled, all those numerous prefixes with unpredictable meaning, irregular forms that even native speakers have problems with (remember, our grammar is highly prescriptive), and many seemingly grammatical categories which are realized not by inflection, but lexically, by derivation. Still, in the indicative mood active voice Russian has 3 tenses (present, past, and future), Ukrainian has 5 (present, past, pluperfect, and 2 future tenses with exactly the same meaning) or 4, pluperfect being practically not used nowadays. And English has 16… Simple, Continuous, Perfect, Perfect Continuous times Present, Past, Future, Future-in-the-Past is 16 tenses. Whyyy? Sooo? Manyyy? What's the purpose of that unbelievable number, if one can well live with 3 or even 2 tenses? How on earth can there be 4 present tenses while I live pretty happily with just 1? Here again, in order to be able to use those 16 tenses one has to learn a new way of analyzing the reality, tracing the actions for continuousness and perfectness (needless to say that continuousness and perfectness are alien to Ukrainian and Russian ways of seeing the reality), and watching closely for a non-past tense not getting accidentally into a sentence with the past in the main clause. The Ukrainian and Russian tenses have relative meanings, and in the subordinate clauses of sentences like “He said he knew me” and “He said he would find me” Ukrainian and Russian use present and future verb forms respectively, so an English learner has to understand the idea of an absolute tenses which is extremely difficult, it's not because the very idea is complicated, but because that idea contradicts all of one's life experience, it's a fundamental difference which is usually learned by many, but understood and used in practice by few. And then, there's that weird question and negation do/does/did thing which is aggravated by the fact that some verbs need it and some don't… Slavic native speakers aren't even usually aware that there's such a thing as a negative form of the verb, you just put the particle ne before the verb, and that's all! (Czechs and Slovaks are, perhaps, aware, since ne is written together with the following verb in their languages). But in English, negating a verb is a complicated thing to do, it comes out that those “negator” don't / doesn't / didn't is inflected for tense, number, and person, very much like it's in Finnish, one would hardly expect such a situation from an Indo-European language. Because of that, the first English tense which is usually presented to a learner is Present Continuous (‘I am doing’), but here again, one faces all these I'm, you're, isn't, aren't, the words made of words of different parts of speech fused together into one, and you have to know when you can use them and when you're to use the full form. And also, there are phrasal verbs, like “go on” or “give up” with unpredictable meaning which compensate for the Slavic prefixed verbs. The English verbals have continuous and perfect forms (e. g. ‘to have been doing’, ‘having done’, etc.) which are not complicated to construct but very complicated to grasp the meaning of, and the way to use them correctly. Then go the modal verbs (“can”, “might”, “must”, and the like) which are not a simple thing in practically all the languages, but in English they are especially complicated, since one is supposed to distinguish the minute details and distinctions of, say, “can do”, “could do”, and “could have done”. I wrote “minute details”, since in Ukrainian and Russian the distinctions among the modal words are very rough. For example, we have the same word for “can” and “may” (U. могти, Ru. мочь) and Russian has the same word for “must”, “should”, “ought to”, “to have to”, “to be to”: должен (actually, there are different words for the latter must-like words, but they are rarely used).
As for the syntax, it's been already mentioned in other answers here, the fixed, rigid word order of English is what makes it complicated. Slavs are used to beginning a sentence with whatever word first comes to one's mind. Subject pronouns are often dropped (Polish, Czech, and Slovak are consistent pro-drop languages). Not so in English, the SVO sequence is practically never broken, which means that to be able to speak English correctly you've got to know what the subject and what the object in the utterance are, that is, you have to learn the sentence analysis, which can be rather difficult, especially for adult learners (children study sentence analysis at school and they usually remember how to do it). Additionally, the ability to find the main clause of the utterance is needed for following the sequence of tenses rule.
Also, English uses different infinitive and participial constructions which are not found at all in Ukrainian and Russian, e. g. “I like her to sing”, “I heard her sing” (oh, no ‘to’ here…), “I heard her singing”, “It turned out to be good”, “I waited for her to sing”, “The rain having stopped we went home”, etc. The correct usage of all those numerous forms of the English verbals is such constructions is a fine art not easily mastered by the majority of learners (not even by the majority of the learners who were persistent enough to reach this point of the curriculum).
And also there are the dialects of English. When I heard Australian English for the first time I couldn't make out anything. For the native speakers of the Slavic languages who are used to long and beautiful words, English words are too short and generally hard to make out in the stream of speech, the English language has a strikingly low level of redundancy as compared with the Slavic languages (cf. American English “I can't take” sounds exactly like “I can take”). Compare a rather typical Ukrainian noun розповсюдження [rɔz.pɔw.ˈsʲu.d͜ʐɛnʲ.nʲa] and its English translation, “spread” (like in ‘spread of HIV’)
In general, the level of analyticity English displays has no parallels among the languages of Europe, and, perhaps, among all the Indo-European languages. “Analytic” is a derivative of “analysis”, and it is deep and complicated analysis of reality that one has to perform in order to be able to use English well.