What is the difference between SEPA and a wire transfer

The term “wire transfer” generally refers to any electronic money transfer mechanism. That includes mechanisms such as SEPA. But in this context, it primarily refers to SWIFT. SWIFT works by partnering banks exchanging payment orders, until the payment reaches the recipient's bank. This is somewhat like packet routing in the internet.

While SWIFT is the standard for international payments, it is subject to various criticism. Fees are unpredictable and high, since each intermediary bank wants to be compensated. The resulting transfers are slow. In the past, the SWIFT architecture has enabled government interference with foreign transactions, such as US surveillance and blocking of intra-EU transfers, or to enforce sanctions on other countries.

We thus see various alternatives to address such shortcomings.

• Within the EU, SEPA has displaced SWIFT almost entirely but is very comparable for users. While SEPA is not necessarily zero-fee, the fees are often negligible and always predictable. Speed depends on the SEPA variant, and varies from seconds to two days.
• Internationally, private companies like Paypal and Transferwise offer transfers with near real-time speed and somewhat cheaper prices than SWIFT.
• In the crypto space, lots of solutions have been proposed. Bitcoin is largely censorship-resistant, but cannot offer real-time speeds and is very expensive (both in direct fees and externalities). Ripple is designed more like a SWIFT replacement, but suffers from its quasi-centralized architecture.

SEPA means "Single Euro Payments Area", and is a direct account-to-account transfer for Euros - exclusively.
Although technically banks would be allowed to charge a fee, in praxis there is no fee charged, except for EU countries that use other currencies (where it is typically conveniently hidden in the exchange ratio).

SEPA Transfer - contrary to wire transfers - take usually one business day (wire transfers typically are same-day).