As the Thieves Guild is a guild, operating like a union, there will be diplomatic and litigious implications of being a member, and identifying yourself as one.
The fact that the Thieves Guild is by this point an officially recognised entity (see Third Century) supports this, and it seems as though the behavior of the guild is in some way sanctioned, if not permitted by the wider authorities of Tamriel.
There could be any number of reasons for this, but Discworld (which shares many parallels with Tamriel!) provides a humorous hypothetical, if not entirely plausible example of what may have happened in Tamriel:
Lord Vetinari realised that what people crave is stability, and that, while it is impossible to stamp out crime altogether, it is possible to regulate it. The major gang leaders of the city were therefore called to the Patrician's Palace, where they agreed to be held responsible for ensuring a socially acceptable number of thefts. While they may have been insincere in this promise, they soon found the Patrician knew too much about them for reneging to be safe.
Given the benefits of unionisation, it could well be that being a member of the Thieves Guild grants further dispensation, such as an accommodated leniency (although I'm not sure if anyone has directly compared sentencing for theft depending on a players allegiance) or one has the option to pay off a guard if caught to let him get away with the goods. (although he doesn't need to wear the armor for this). Still, just a few possibilities of why it would benefit one to be a member of the Guild.
The outfit also has a practical function, as others have commented.
It could be easier to consider the Thieves Guild in comparison to modern special forces, or perhaps more ominously 'private security' corporate mercenaries like Blackwater.
This is an organisation that widely disperses the fact that they are not bound by the same Rules of Engagement (or 'laws') that official military units do, and as such are free to 'break the law' without the same recrimination as an official authority. They even have their own insignia, and they are not perturbed in any way by their reputation for false/illegal conduct. Regardless, they are able to operate as long as they are not caught red handed: they are, like the Thieves Guild, tolerated but unliked.
The comparison between the organisations can be drawn further:
It would seem foolish to identify oneself openly as part of a scorned and widely reviled organisation, but it sends a clear signal to anyone who would seek to cause undue or unwarranted harm to a member: an assailant attacks not just an individual, but an entire organisation in the process. Anyone whose actions are found to be unsubstantiated can expect to feel the full weight of the organisation in retribution, which must be intimidating.
This will extend to their political and diplomatic influence as well: A guard who falsely imprisons a Thieves Guild member could find himself demoted for causing an incident, possibly interfering in a sanctioned theft: like a bailiff.
The diplomacy of Tamriel must be very complicated.