An institute is defined as a society or organization having a particular objective, whether a scientific, educational or social one. A technical college, on the other hand, is more specific. It's defined as a college providing courses in a range of practical subjects, such as information technology, applied sciences, engineering, agriculture and secretarial skills. So, by definition, an institute has a more general connotation, having a potential reference to organizations that don't provide formal education.
Institutes that do provide formal education still differ from technical colleges in at least two ways. One is prestige. When you think of the Massachusetts Institute of Technology (web.mit.edu), or as it's more commonly known, MIT, there's a hint of prestige attached, largely due to its reputation housing high caliber students and professors, and in producing formidable research. But when you think of a school such as the Los Angeles Trade-Technical College (lattc.edu), the level of prestige goes down. A report published by the annual "Times Higher Education" ranked 100 schools according to their reputation. MIT ranked second. The California Institute of Technology and the Georgia Institute of Technology were also on the list. But nowhere on the list was a technical college.
You might argue that the aforementioned list covers universities, and so that's the reason why technical colleges weren't included. But that just further emphasizes the point — institutes, because of their prestige, are classified alongside universities. Technical colleges are not. This isn't to say that technical colleges don't provide quality education. A study by research company Belden Russonello and Stewart, LLC, showed that many laud technical colleges for their empowering education and affordable tuition.
While, in general, institutes tend to have a better reputation than technical colleges, sometimes the two designations are used interchangeably. For example, ITT Technical Institute (itt-tech.edu), while an institute by name, is described on its website as a leading private college focused on technology-oriented study programs. So the difference between an institute and technical college, sometimes, may simply be nominal. Investigate whatever school you're considering to see which is the better fit for your situation and goals — educationally, financially and in terms of reputation.