Yes, you may very well qualify for tradename protection, and it's quite possible that you have a viable potential lawsuit. Your legal remedies may include a right to recover money damages to compensate you for your lost business or damage to your reputation, and possibly a right to a court order forbidding your competitor from continuing to do business under the infringing name.
There's not much that is black and white in trademark/tradename law, however, and these cases are highly dependent on their individual facts.
Thus, to answer your question with a higher degree of confidence, it would be useful to know, at a minimum, the exact name of your business and that of your new competitor; what kind of business it is; how long you've done business under that name; and what efforts you and your new competitor have undertaken to advertise or otherwise promote your businesses. There are certain kinds of business names that are subject to greater tradename protection than others, and there are certain kinds that are subject to less. If, for example, you've been doing business as "Bob Smith Roofing" and your name is Bob Smith, and your competitor is doing business now as "Rick Smith Roofing" because HIS name really is Rick Smith, you may have no effective remedy.
But the likelihood of consumer confusion is very important in these cases, and the fact that you have existing customers who appear to be at risk of being misled as to whom they're doing business with is helpful to your potential legal claims.
Sitting on your rights -- delaying in making a formal complaint and demand, or then delaying in seeking to enforce your legal remedies in court -- can quickly destroy them. If you don't take prompt action to protect your rights, in other words, you may later be barred from enforcing them at all.
You're likely to need more detailed advice -- in connection with which you can communicate under the privacy cloak of attorney-client privilege -- than you can get on an internet website. Consult a lawyer. You certainly should look for someone with some prior experience in tradename/trademark cases. And indeed, there are specialists who do almost nothing BUT "intellectual property cases," but in my personal opinion, those specialists -- with some exceptions -- tend to be very highly priced, may in fact have less experience in slugging things out in court, and sometimes they end up being less cost-effective. These cases can be surprisingly expensive for both sides, racking up enormous amounts of attorneys fees in a startlingly short time, so it's important to find someone whose services are indeed cost-effective.
It's possible that you may be able to frame your claims in a way that would permit you, if you won, to also recovery an award of reasonable attorneys' fees. But unless your case is very, very strong indeed and the potential defendant is very solvent, however, it's likely that most potential counsel are only going to be willing to undertake your representation on a regular hourly fee basis -- not on a contingent fee basis (where the lawyer is only compensated out of the proceeds of any judgment or settlement you collect).
A business can have any or all of the following, and they can be the same or not: 1) an entity name, if your business is organized as a corporation or LLC or with a ficitious business name for an individual, partnership, or other entity 2) a domain name, where it does business on the internet at a unque address, 3) a trademark, acquired by use when used to identify the source of goods or services, and trademarkable under both state and federal law.
The Secretary of Stait in your state is the one who prevents two comapnies from using the same entity name. The domain name duplication is governed by ICANN, which all interent domain name registrars require their customers to subscribe to, and which provides a process to resolve disputes where someone is "cybersquatting" on someone else's website or using their domain name so as to infringe someone else's trademark without a legitimate reason.
As for trademark rights, no one on Avvo can answer an infringement question with much more than "it depends on the facts." As my colleague poinbted out, the fact that you're getting feedback that your customers are asking about you moving, because of this other business, is good evidence of confusion, and the trademark rules are designed to protect consumers -- not trademark rights holders --from misleading trademarks that confuse them about the actual source of good and sevrices.
You haven't provided any information about whether the name you're using is protectible (for example, descriptive and generic names like "Houston Dry Cleaners," is a very weak mark, if it would get registration at all), whether you're using a business name as trademark, and if so, for how long, and how, and the same questions apply to your competitor. You need to see an IP lawyer to disclose everything and get some advice specific to your particular situation.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
You've already received excellent information from both Mr. Dyer and Ms. Koslyn.
My only input is to respectfully -- though strongly -- disagree with Mr. Dyer's suggestion that intellectual property litigators may not be worth hiring because of their perceived higher rates.
I've been involved in too many cases where generalist trial counsel simply did not have the knowledge base required to adequately (which means quickly and efficiently) think through the best way to resolve trademark disputes. Far too often a statute (federal or state) or common law rule is ignored, far too often the cease and desist letter is either impotent, improperly tailored, or worded such that it triggers a declaratory relief lawsuit from the recipient, and far too often the complaint is very badly structured and legally and factually insufficient. Often, there is no mechanism to fix these mistakes.
If a thing is worth doing, it is worth doing right.
One other note: because trade name / trademark law is a mix of both state and federal law it almost always makes sense to hire an intellectual property litigator who's registered to practice in the aggrieved party's home state.