First of all, as a Dutch speaker I can understand written German fairly well. If German speakers speak slowly and with basic vocabulary, then I can understand spoken German as well. I studied German for two years in school, so I have some knowledge of the language. However I do not claim that I can speak it. I’m planning to do an erasmus next year to Germany.
To give you an example of a grammatical difference between the two languages:
Dutch is kind of simplified compared to German, since there are no cases (only the nominative and accusative, just like in English). There are still three genders: the male/female article is ‘DE’, the neuter article is ‘HET’. In most cases the genders are similar in German and Dutch, though naturally there are a lot of exceptions.
The declensions of the adjectives are very different from German.
For instance: the word ‘mooi’ means ‘beautiful’, so we could say ‘het meisje is mooi’, which means ‘the girl is beautiful’. ‘het mooiE meisje’ means ‘the beautiful girl’, but when you write ‘een mooi meisje’, which means ‘a beautiful girl’, you write ‘mooi’ instead of ‘mooie’.
Similarly: ‘de man is mooi’, ‘de mooiE man’, but ‘de mooiE man’.
I have spoken with many foreigners, even with people who have mastered Dutch at a very high level, but they still make many mistakes to this. In German this is more complicated, because of all the cases, and you don’t just add ‘e’ to the stem, but depending on the case ‘es, en, …’.
The verb declensions are different. Some verbs have different meanings too, the number one example is: the German verb’mögen’ vs. the Dutch verb ‘mogen’. In German ‘mögen’ means ‘to like’, while in Dutch ‘mogen’ means ‘to be allowed’.
The different tenses are very similar, though the conjugations and irregular verbs are different (past tense, past participle, etc. - you will have to study them).
Another very difficult item in Dutch is the word ‘er’. If you are interested in hearing more about this, just ask me. This structure is not used in German like it is in Dutch. It’s really complicated where to put it in a sentence, because mostly for this word there is one fixed place (and foreigners sometimes put it somewhere else, so that it won’t sound natural anymore).
The last thing I’d want to tell you about it the similarity between the German and Dutch separable verbs (which is also very difficult without the knowledge of German). For instance, in Dutch we have the word vóórkomen (1) and voorkómen (2).
(1) Ik voorkom = I avert/prevent
(2) Het komt voor = It occurs
more info here: voorkomen - Wiktionary
More questions? Maybe if you have some more specific ones, just ask me.