Verbs and verb forms in American and British English grammar:

You are right, Anwar, there are not very many differences in grammar. Rather more when it comes to vocabulary and idiom.

Many of the differences in grammar relate to choice of verb or verb form. Here are some of the most common:
 

time adverbs with past simple/present perfect

Past-time adverbs, such as just, ever, already and yet are often used with the past simple in American English, whereas in British English they would normally be used with the present perfect. Compare the following:

• Did you phone her yet?
• Have you phoned her yet?
  
  
• Did you eat already?
• Have you already eaten?
  
  
• Garry? You missed him. He just left.
• Garry? You've missed him. He's just left.
  
  
• Did you ever go to Canada?
• Have you ever been to Canada?

Do you have...? / Have you got...?

In all varieties of English, the 'do' forms of have are used to express habit or repetition:

• Do you always have fruit and cereal for breakfast?
• Do you sometimes have a shower in the morning when you wake up?

In American English, the 'do' forms of have are commonly used when referring to particular situations. In British English, we often prefer have with got in these contexts. Compare the following:

• Do you have time to finish this report before you leave?
• Have you got time to finish this report before you leave?
  
  
• Do you have a problem with this?
• Have you got a problem with this?

In American English, got and do forms are often mixed. In British English, they would not be:

• We've got a new car! ~ You do?
• We've got a new car! ~ You have?

regular and irregular past tenses and past participles

The following verbs are regular in American English, but are often irregular with -t rather than -ed in British English:

burn
dream
learn
smell
spill
spoil

• The kitchen smelled of roast chicken. Dinner was ready.
• The kitchen smelt of roast chicken, Dinner was ready.
• I have learned that it is better to be safe than sorry.
• I have learnt that it is better to be safe than sorry.
• He had spoiled his paper by spilling his coffee on it.
• He had spoilt his paper by spilling his coffee on it.
  

The following verbs are regular in British English, but irregular in American English:

dive
fit
wet

• All her clothes fit into the suitcase.
• All her clothes fitted into the suitcase.
  
  
• She wet her long blond hair before pushing it under her bathing cap.
• She wetted her long hair before pushing it under her bathing cap.
  
  
• Then she dove into the pool with all her clothes on.
• Then she dived into the pool with all her clothes on.
  

Can / could with verbs of perception

In British English, we normally use can or could with verbs of perception such as see, hear, taste, feel, smell, when American English will often use these verbs independently of can or could. Compare the following:

• When I went into the garden, I could smell the cherry wood burning on the camp fire.
• When I went into the garden, I smelled the cherry wood burning on the camp fire.
  
  
• I could hear Caroline approaching through the long grass.
• I heard Caroline approaching through the long grass.

going to / gonna

In talking about plans and intentions, going to is often replaced by gonna in informal speech, especially in American English. Compare the following:

• We'll see you at the game. You're gonna play, right?
• We'll see you at the game. You're going to play, aren't you?

And as they say in American English, and now increasingly in British English:

... have a nice day!