Could you please help me to understand differences .
Want to take a taxi? (has "to")
We'd better take a bus. (no "to")
These two sentences indeed carry different meaning.
'Want to take a taxi?' is an invitation to ask somebody if they want to take a taxi or not. It might be a friend who is saying this, or the taxi driver.
'We'd better take a bus' is just another version of 'We had better take a bus ', where 'we had' is shortened into 'we'd'.
The formation of 'Had better (without to) infinitive' is used to give advice about a specific situation. Thus, in your example, the speaker is giving an advice or suggestion, instead of giving an invitation.
e.g. You'd better drink some water --> The speaker is advising 'you' to drink more water, so perhaps 'you' is looking very thirsty and exhausted
Want to drink some water? --> The speaker is inviting 'you' to drink some water, so you can imagine the speaker might be handing out his water bottle to 'you'
For the question of whether or not to use 'to' here, 'to' is in the first sentence because 'want to + verb' is used to express a desire of something.
e.g. 'I want to go swimming'.
But! This only works when it's a verb. We don't use 'to' when it's a noun!
e.g. I want attention.
The second sentence does not employ 'to' as it's using the formation of 'had better + infinitive', so adding 'to' would break the formation.
Hope this helps!