Is it recommended to do the Sunnah prayers all the time, or is it better not to do them sometimes so that they do not become like the obligatory prayers?

Praise
be to Allah.

Firstly: 

Always doing that which Allah has made obligatory is one of
the duties of Islam, and is the foundation of steadfastness on the straight
path of Allah. Always doing the regular Sunnah prayers in particular, and
virtuous deeds in general, is something that is recommended in Islamic
teaching. The Prophet (blessings and peace of Allah be upon him) was asked:
Which deed is dearest to Allah? He said: “That which is done persistently,
even if it is little.” Narrated by al-Bukhaari (6465) and Muslim (782).

 Al-Bukhaari (1987) and Muslim (783) narrated from ‘Alqamah:
I said to ‘Aa’ishah (may Allah be pleased with her): Did the Messenger of
Allah (blessings and peace of Allah be upon him) single out any particular
day for anything? She said: No, his actions were persistent, and who of you
can do what the Messenger of Allah (blessings and peace of Allah be upon
him) could do? 

An-Nawawi (may Allah have mercy on him) said: 

The words “his actions were persistent” mean that he would
always do them and not cease to do them. 

Al-Haafiz (may Allah have mercy on him) said:

 The linguists said that the word daymah (translated above as
persistent) originally referred to rain that lasts for several days, then
the word came to be used to refer to anything that is ongoing. 

Al-Bukhaari (6461) and Muslim (741) narrated that Masrooq
said: I asked ‘Aa’ishah (may Allah be pleased with her): which deed was most
beloved to the Prophet (blessings and peace of Allah be upon him)? She said:
That which is done persistently. 

Muslim (746) narrated that ‘Aa’ishah said: If the Messenger
of Allah (blessings and peace of Allah be upon him) did an action, he would
persist in it. If he slept at night or was sick, he would still pray twelve
rak‘ahs during the day. 

Thus it is clear that there is nothing wrong with
persistently doing Sunnah and naafil righteous deeds, in fact that is
recommended and encouraged, because that was what the Prophet (blessings and
peace of Allah be upon him) did and taught others to do, and he encouraged
others to do that in more than one hadith, such as that which was narrated
by at-Tirmidhi (414) from ‘Aa’ishah (may Allah be pleased with her),
according to which the Prophet (blessings and peace of Allah be upon him)
said: “Whoever persists in praying twelve rak‘ahs of Sunnah prayer, Allah
will build for him a house in Paradise.” Classed as saheeh by Shaykh al-Albaani
in Saheeh at-Tirmidhi. 

Abu Dawood (1269) and at-Tirmidhi (428) narrated that Umm
Habeebah, the wife of the Prophet (blessings and peace of Allah be upon him)
said: I heard the Messenger of Allah (blessings and peace of Allah be upon
him) say: “Whoever persists in praying four rak‘ahs before Zuhr and four
rak‘ahs after it, Allah will forbid him to the Fire.”

Classed as saheeh aby al-Albaani in Saheeh Abi Dawood. 

At-Tirmidhi (3410) narrated – and classed it as saheeh – that
‘Abdullah ibn ‘Amr (may Allah be pleased with him) said: The Messenger of
Allah (blessings and peace of Allah be upon him) said: “There are two deeds
that no Muslim man persists in doing but he will enter Paradise. Indeed they
are easy but those who do them are few: glorifying Allah (by saying Subhaan
Allah) following every prayer ten times, praising Him (by saying al-hamdu
Lillah) ten times, and magnifying Him (by saying Allahu akbar) ten times.
That is one hundred and fifty on the lips and one thousand and five hundred
in the balance. And when you go to bed, then glorify Him, magnify Him and
praise Him one hundred times; that will be one hundred on the lips and one
thousand in the balance. And who among you could do in one day and night two
thousand and five hundred bad deeds?” They said: How could anyone fail to do
that? He said: “The Shaytaan comes to one of you when he is praying and
says: Remember such and such, remember such and such, so that when he
finishes his prayer, he may forget to do that; or he comes to him when he is
in his bed and keeps on making him sleepy until he falls asleep. 

It was also narrated by Abu Dawood (5065) as follows: “There
are two deeds that a Muslim does not do regularly but he will enter
Paradise. They are easy but those who do them are few…

Classed as saheeh by al-Albaani in Saheeh at-Tirmidhi.                     

Al-Mubaarakfoori said in Tuhfat al-Ahwadhi: 

“Those who do them are few” i.e., those who do them
consistently. 

Secondly: 

With regard to them not being like the obligatory prayers,
this is clear, and it is sufficient for the individual to be aware of that
and to intend them as naafil whilst doing them, and make that clear to
people, if he is explaining the Sunnah to people. We are not now living at a
time in which the teachings of Islam were fresh and new, where it can be
imagined that people could confuse obligatory deeds with naafil ones.

Moreover, it will inevitably happen that a person may be
interrupted by urgent work, travel, sickness and the like. In such cases
there will be stored up for him that deed that he stopped doing, and he
should not deliberately omit it because of what is mentioned above. 

Thirdly: 

With regard to that which it is narrated in the Sunnah that
the Prophet (blessings and peace of Allah be upon him) did it sometimes and
did not do it sometimes, if this is how it was done by the Prophet
(blessings and peace of Allah be upon him), one should pay attention to that
by doing that thing sometimes and not doing it sometimes, because that is
preferable to doing it constantly.

 An example of that is sometimes reciting in the last two
rak‘ahs of Zuhr and ‘Asr prayer something in addition to al-Faatihah, and so
on. 

And Allah knows best.