They are different.
A Tzaddik is someone who does everything correctly (see Mesilat Yisharim, Chapters about Nekiyut - Chapter 13 clarifies this is referring to tzaddik).
A Chassid is someone who does everything correctly, and his intentions are pure, meaning he doesn't do it for his own benefit at all, but does it purely for the sake of Hashem (see Mesilat Yisharim chapters on Chassid)
This certainly applies to non-Jews as well. One way to look at it is like this. A non-Jew, without a Torah, can be an extremely nice guy, naturally. As a result, he may indeed be keeping all the laws, out of his own natural, logical niceness. That won't cut it, he needs to do it because Hashem said so for it to be considered something "above and beyond". Or another way to say it, in the world to come, Hashem will be fully revealed and therefore it is important that a non-Jew forms a personal relationship with Hashem because the real reward of the world to come is enjoying relationship with Hashem. The former "tzaddikei umot haolam" have no relationship with Hashem, they are just doing good things, which will also be, tit for tat, be rewarded with "things" i.e. rewarded in this world, not the next, which is a world "above things", where the spiritual totally subdues the physical.
EDIT: Here is the tosefta in english (my own translation, but I used this to aid translation):
It is taught that the sons of the wicked of Israel don't have a share in the world to come, and will not live in the world to come, as it is written "For lo! That day is at hand, burning like an oven". These are the words of Rabbanan Gamliel. Rabbi Yehoshua says they will come to the World to Come, and furthermore, he says "God protects the simple" (a Talmudic principle that God doesn't inflict His punishments on people who couldn't have known better), and furthermore "Hew down the tree, lop off its branches...But leave the stump with its roots in the ground". Rabbanan Gamliel says, what about "shall burn them to ashes and leave of them neither stock nor boughs.", which means that Hashem will not leave a mitzva or parts of a mitzva to be rewarded in the World to Come. Another explanation is that "root" means soul and "branch" means body.
The children of the wicked of the gentiles will not live in the World
to Come, nor shall they be judged. Rebbi Eliezer holds: none of the
gentiles have a share in the world to come, for it is written "the
wicked will return to sheol, all the nations that forget God". "The
wicked shall return to sheol" - these are the wicked in Israel. "All
the nations that forget God" - these are the wicked among the
gentiles. Rabbi Yehoshua said to him: If the pasuk had said "the
wicked shall return to sheol, all the nations...", and then said no
more, I would have agreed with you that it is talking about the wicked
of Israel and the wicked of the gentiles, but since the pasuk says
"who forget God", there must be righteous (tzaddikim) men among the
gentiles who have a share in the world to come.
The school of Shammai say there are three categories: one for eternal
life in the world to come, one for shame and everlasting contempt, the
former are accounted wholly righteous, and the latter are accounted
wholly wicked, and a third category that descend to gehimon and scream
and return and are healed as it is written "I will bring the third
part through the fire...", and Chana said about them "Hashem kills and
Hashem gives life...".
Beit Hillel says (on the third category) "He is great in mercy", which
means He leans in the direction of mercy, and of them David said "I
love it that Hashem listened...", and of them this whole psalm is
written.
The rebels of Israel and the rebels of the gentiles descend to gehinom
with their bodies, and are judged for 12 months, and after twelve
months their souls are destroyed and their bodies burnt. Gehinom casts
them forth and they become dust, the wind blows them around and
scatters them under the soles of the feet of the righteous, as it is
written: "And they shall tread on the wicked, for they shall be
dust...". But the conspirators, the heretics and the deniers of Torah,
and the abandoners of the ways of the community, and those who don't
believe in the resurrection, and those that sin and lead the masses to
sin like Yeravam and Avihu, and those who struck terror in the land of
the living, and those who stretched their hand against the Temple;
gehinom is closed for them, and they are judged for all generations,
as it is written "And they shall go forth and look at the corpses of
the men...". Sheol will be destroyed but they will not, for it is
written "And their form will cause sheol..." What caused this on them?
Because they stretched their hand against the Temple as it is written
"because of Zvul" and Zvul means Temple as it is written "I have
surely built the house of Zvul for You".
This Tosefta seems to be matching up exactly with what I said above (which, by the way, is based on the Lubavitcher Rebbe's teaching, who was of course an expert in all of Torah, including Tosefta and Rambam).
There seem to be three categories: the wicked gentiles, the standard gentiles, and the "righteous" gentiles.
Firstly, don't worry about the terminology. The word "tzaddik" here is not in contrast to "chassid", as Rabbi Yehoshua seems to recognise that there are 3 categories, and the defining difference between the middle category and the latter is "those who forget God", not how many good/bad deeds they have. Nay, the middle category could have nothing but good deeds their whole life, in which case they would certainly be considered tzaddikim, in the more general sense of the balance between their good deeds and bad deeds, but they don't remember/involve God.
Rabbi Yehoshua is arguing with Rebbi Eliezer's idea that no gentiles are going to be in the world to come. His proof is that those who didn't forget God are spoken about, implying there are some gentiles who didn't forget God (Rambam's "chassidei umot olam"), i.e. do their good deeds because He said so, and they certainly have a share in the world to come. The fact that no more is said implies that this argument was accepted (but not definitively).
