"to phone" vs "to phone up"
Is there any difference in meaning between just "to phone" and the phrasal verb "to phone up"?
For example, do the sentences a and b be below have absolutely the same meaning?

1a John has just phoned to say that he'll be late.
1b John has just phoned up to say that he'll be late.

2a I phoned the police and told them about it, but they said there was nothing they could do.
2b I phoned up the police and told them about it, but they said there was nothing they could do.

Question

"to phone" vs "to phone up"
Is there any difference in meaning between just "to phone" and the phrasal verb "to phone up"? 
For example, do the sentences a and b be below have absolutely the same meaning?

1a John has just phoned to say that he'll be late.
1b John has just phoned up to say that he'll be late.

2a I phoned the police and told them about it, but they said there was nothing they could do.
2b I phoned up the police and told them about it, but they said there was nothing they could do.

Answer

Hey Ksenia,

I should say that all of your sentences sound rather good. Besides, you are right: both the verb ''to phone'' and the phrasal verb ''to phone up'' have pretty much the same meaning.

1) 'To phone' = To make or try to make a telephone connection with;
2) ''To phone up'' = to (try to) speak to (someone) by means of the telephone.
E.g.: ''I need to phone my mother.''
E.g.: ''My friend phoned us to say he had arrived.
E.g.: ''You should phone up your lawyer and tell him everything that happened.''
E.g.: ''Has he phoned up his sister today?''

* HOWEVER, you should know that the usage of the verb ''phone'' is much more common than that of the phrasal verb ''phone up''. Moreover, you usually use the former when you want to make a telephone connection with a 'place' (the latter is not commonly used in that sense).

E.g.: ''He phoned work to tell them he was ill.''
E.g.: ''I'll phone the cinema and find out what time the film starts.''

* NOTICE: Do not use 'to' after phone. Don't say, for example, 'I need to phone to my mother'.
Did you get the idea?

Hope that helps!!!
Should you have any further doubts, do not hesitate to contact me. I shall be pleased to help.

Answer

Question 1: diffence of meaning (of up)

Answer: 'up' as an adverb used following a verb means

(a)  to a disired or better situation,  
(b) to an end.

tidy up/wrap up/phone up
line up/set up

In sense (b), verbs collocate with different adverbs: up/out/off, sometimes two of them are interchangeable.

When you use 'up', relative to not using it, you put a slight emphasis onto your meanig.

Question 2:  when to and when not to use 'up'?

For a certain sense of some verbs ,  the 'up' is optional. There seems to be no obvious regularity which verbs can(not) be used with 'up' , or rather if you want to give a stress, you add 'up' to them.

On the contrary, for some other verbs, the verb alone and the struction of ' verb + up' may be different words, with the latter called a 'phrasal verb', for example, set vs set up.

Answer

This must be UK English. American English we will use "call" instead of "Phone."