Jesus is accusing the people of his time of supposedly believing what Moses wrote, but not believing what he says and does. We may look at this through the eyes of the New Covenant and think "how ignorant of them – it’s so obvious!" However, the message still applies to us today, especially regarding the theological differences between Catholics and Protestants.
Some Protestants (not all, but some that are Fundamentalists) believe in a "literal" interpretation of the Bible. This can be understood to mean that they believe that the universe was literally created in 6 days. The Bible says so, therefore it is so. This is how they can be seen to be claiming to believe Moses – after all, according to some, Moses wrote the first 5 books of the Bible. However, with all the literal interpretations of passages from the Bible, both Old and New Testaments, there is one section that they refuse to read literally: John 6:26-71. In these verses (almost a full chapter), Jesus says explicitly many times that they are to eat his flesh and drink his blood. Yet, the people who interpret the Bible literally will come up with many different explanations of how Jesus meant "unless you eat the flesh of the Son of Man and drink his blood, you do not have life within you," symbolically, or that, "my flesh is true food, and my blood is true drink" is to be taken metaphorically. They claim to believe Moses, yet do not believe Jesus.
This type of double-standard is not just held by literalist Protestants – there are a large number of Catholics who think this way too, although their attitude is not toward Moses and Jesus, but toward Jesus and the Church. There are many Catholics who can be called "cultural Catholics", because they were born into a Catholic family and follow some customs (maybe even go to Church on some or even most Sundays), but don’t really take much of it to heart. Then there are those that are called "Catholics-in-name-only" or CINOs – they check off a box on a census form that says they are Catholic, but you’re lucky if you see them in a congregation any time other than Easter or Christmas. (They are also known as the "lily and poinsettia crowd".) And finally you have the "Cafeteria Catholics" – these are the ones who pick and choose what they will and won’t accept or follow as teachings of the Catholic Church, most commonly on the issues of abortion, contraception, divorce, and papal infallibility (see The Curt Jester’s "The Cafeteria is Now Open" for more menu items).
All of these "Catholics" are guilty of the same line of thinking, to which the Church (or any member willing to speak up for it) could easily reply as follows:
Do not think that the Church will accuse you before the Father: the one who will accuse you is Jesus, in whom you have placed your hope. For if you had believed Jesus, you would have believed His Church, because He talked about it. But if you do not believe his sayings, how will you believe the Church’s teachings? (cf. John 5:45-47)
This does not mean we are to reject them outright when they do come to Mass. Jesus is the mediator who stands in the gap for all of us, when God the Father would destroy us for our depravity and our lack of faith. The church walls do not come crumbling down on us every time some sinner comes back to the fold because we are a Church of sinners. None of us is without fault – we are just trying to follow all the teachings of the Church, stumbling and straggling along the way. The Church was not made for the healthy – it was made for the sick who want to be healthy.
Pray for those you know that fall into any of the groups mentioned above, and pray for God’s mercy upon them. Pray also for yourself, because most likely you once wore one of those labels too, and we all need God’s grace so they we may run and not grow weary and end up falling back into one of those categories.