As someone who was born into Orthodox Christian faith, has been baptized, attended church occasionally, has read full Bible, both New and Old Testament (BTW the Orthodox version of which includes 11 more "books" than the Protestant, 4 more than the Catholic), who has high respect for the teaching of Christ, appreciation of Father, and deep connection with Holy Spirit, and also as one who's been practicing and studying Buddhism for decades now, and became independent of others in regards to Buddha's message -- I suppose I'm in a good position to answer this question.
The words representing "faith" (śraddhā & pistis) in both religions share most of their semantic fields: belief, trust, confidence, loyalty. Sraddha includes a couple more meanings like assent/approval/consent/reverence, and appetite/longing/wish/expectation -- which too are connotations of "faith" as it is understood in Christianity. So overall, we have two rather similar words.
Faith in Early Christianity
The "main" definition of faith in Christianity is found in Letter to the Hebrews by the Apostle Paul and goes like this:
As for faith, it is realization of the expected and confidence in the invisible.
In Orthodox Christianity this phrase is explained as follows: "Not only we who carry Christ's name consider the faith as the greatest, but everything done in the world, even by people strange to the Church, is done by faith, since the one who does not believe that [he] will rip the grown fruits, will not tolerate the labors". Another example given is the faith of the seafarers, who "entrust their lives to a small wood" and "give themselves to unknown hopes and have by themselves only faith, which for them is more reliable than any anchor".
From this we can see that "confidence in the invisible" is not as much blind faith in the existence of something we can't verify, but rather "realization of the expected" -- confidence in the outcome of our effort, based on our prior experience and our knowledge of how the world operates. This means that as Christians interacts with the world, they are supposed to get first-hand evidence/support of why the Son's Commandments make sense, see Holy Spirit at work, and recognize the law of Father behind all.
At the same time Orthodox Christians, like Zen Buddhists, recognize that our understanding can never exhaust the depth of things. There will always remain something which escapes analysis and which can not be expressed in concepts. Therefore, any realistic view of the world necessarily includes reverent recognition of this unknowable depth of things, that which constitutes their true, indefinable essence. This appreciation of the ultimate ineffability of the Law is a major component of faith.
Standing on these two legs -- appreciation of the order of things, and fascination with the depths -- we can have courage to do what's right, even though there is no 100% explicit guarantee it will work. This is faith.
Faith in later Christianity
Where this starts breaking up, is in its application to actual Christian practice, which virtually demands a Christian to take the following creed:
I believe (
pisteuo) in God,
the Father almighty,
Creator of heaven and earth,
and in Jesus Christ, his only Son, our Lord,
who was conceived by the Holy Spirit,
born of the Virgin Mary,
suffered under Pontius Pilate,
was crucified, died and was buried;
he descended into hell;
on the third day he rose again from the dead;
he ascended into heaven,
and is seated at the right hand of God the Father almighty;
from there he will come to judge the living and the dead.
I believe in the Holy Spirit,
the holy catholic Church,
the communion of saints,
the forgiveness of sins,
the resurrection of the body,
and life everlasting. Amen.
So as we can see, what started as very rational declaration of faith as expression of experience, understanding, gratitude, fascination, courage, and righteousness -- was later turned into an ultimatum.
Faith in Early Buddhism
Although many Buddhist define their faith as a growing conviction one gains as one is exposed to Dharma and progressively verifies it on one's own experience, this is not what I see in Pali Canon.
Instead, Buddha speaks of faith as the first phase of the path, during which the practitioner gets inspired by the message of Buddha, based solely on a verbal exposure (MN 125):
A householder or a householder's son or one born in another family hears that dhamma. Having heard that dhamma he gains faith in the Tathagata. Endowed with this faith that he has acquired, he reflects [on his life and decides to joins the spiritual path]
At this point one is called faith-follower (SN 25.3):
 One who has conviction & belief that these phenomena are this way is called a faith-follower: one who has entered the orderliness of rightness, entered the plane of people of integrity, transcended the plane of the run-of-the-mill.
The following two phases will help us understand the limits of faith:
 One who, after pondering with a modicum of discernment, has accepted that these phenomena are this way is called a Dhamma-follower.
 One who knows and sees that these phenomena are this way is called a stream-enterer.
Such unwavering faith, that drives one until one sees for himself, is often mentioned in Pali Canon in context of the factors of the path, along with persistence etc. (MN 70):
For a disciple who has faith in the Teacher's message & lives to penetrate it, what accords with the Dhamma is this: 'The Blessed One is the Teacher, I am a disciple. He is the one who knows, not I'... 'Gladly would I let the flesh & blood in my body dry up, leaving just the skin, tendons, & bones, but if I have not attained what can be reached through human firmness, human persistence, human striving, there will be no relaxing my persistence.'
Faith in later Buddhism
By the time of Vasubandhu's Abhidharmakosa (4th or 5th century CE), faith is beginning to acquire a character of dogma:
If it is asked, what is faith? Faith is full confidence in the efficacy of actions and their results (i.e. karma), in the (noble) truths, and in the Three Jewels; it is also an aspiration (for spiritual attainment) and clearminded appreciation (of the truth of Dharma).
This eventually becomes e.g. a following dogma:
How is faith perfect?
This is believing deeply that the Buddha, Dharma and Sangha are Eternal,
that all Buddhas of the ten directions make use of
and that beings and
icchantikas all possess the
It is not believing that the Tathagata is subject to birth, old age, illness, and death,
that he has undergone austerities,
and that Devadatta really caused blood to flow from the Buddha's body,
that the Tathagata ultimately enters Nirvana,
and that authentic Dharma dies out.
Reminds something, huh? :)
In both Christianity and Buddhism we can see the process of gradual codification / dogmatization of symbols and principles, this is normal. Both religions have layers, and if you explore the populations of practicing representatives of both religions, you will always find a gamut of sophistication, from the most superficial to the most profound.
Putting aside the question of the cyclic process of regression and reinstitution of Sat-Dharma (True Teaching), we can see that originally the faith in Buddhism and the faith in Christianity were kinda opposite. Indeed, like I showed above, in Christianity faith used to be an expression of one's (however limited) insight into the Cosmic Law, first and foremost in its ethical aspect. While in Buddhism faith used to refer to a condition of inspiration and unwavering drive one has until one confirms the teaching first-hand. The object of Christian faith, although implicit, is here, while the object of Buddhist faith is over there.
This is because early Buddhism is based on a metaphor of Ground, Path, and Fruition -- while Christianity is in some sense based on a metaphor of Field (whether the battlefield of good and evil, or the field of one's life). The later Buddhism, esp. some strands of Mahayana have adopted this notion of Law (Tao, Bup etc.) and became somewhat less goal-oriented, which found reflection in their definition of faith.
To finish this up, let's consider a perpetual question: which one is better? In my sincere opinion, both the faith as appreciation of the implicit and ultimately inexpressible but evidently effective Law, and the faith as putting one's trust in not yet clear but inspiring the confidence Teaching are important. One starts with the second type and eventually attains the first type. Could we go as far as to say that Buddhist path culminates in first-hand knowledge of the Law of God? May be. This is how I see it anyway ;)