Here are six select differences between the systems which is hardly comprehensive.
1. Taking of evidence, finality, and scope of appeal.
In common law systems, evidence is usually taken in a consolidated single trial, after which a decision on the merits of the case is made. There are fairly strict rules of evidence regarding what can be considered.
The factual determinations made at that trial are final. No new evidence can be received on appeal and generally speaking an appellate court can't second guess interpretation of the evidence presented (such as credibility determinations) made in the trial court on appeal. There is, however, a verbatim record of what happened in the trial court for an appellate court to review.
In civil law systems, evidence is usually taken piecemeal over multiple hearings at which only a small number of the total number of witnesses in a case are presented. There are fairly loose rules of evidence. Detailed notes are taken of evidentiary hearings by the judge(s), but a verbatim record of the trial court proceedings is usually not taken.
On a first direct appeal of a first instance court decision, the appellate court can review both legal and factual determinations made in the first instance court and can receive new evidence in a partial trial de novo of the case.
In the U.S., in a very large share of civil cases and in all criminal cases, there is a right to trial by jury. This right is actually exercised in the vast majority of serious criminal cases and personal injury cases and fraud cases, and in a large share of less serious criminal cases and breach of contract cases.
In most other common law legal systems, jury trials are available in civil cases only in a handful of exceptional circumstances that probably make up less than 1% of civil cases, and in serious criminal cases.
In a jury trial, a single judge presides of pre-trial proceedings, and orchestrates the jury trial itself such as overseeing jury selection, ruling on evidence objections, and instructing the jury prior to its deliberations regarding the applicable law. The jury's verdict is generally only the bottom line result with no explanation and the means by which it came to its conclusion are secret.
In civil law systems, all civil cases and all or all but the most serious criminal cases are decided by judges only. Minor criminal offenses and civil cases with a small amount in controversy are heard by a single judge and appealed to a panel of three more senior judges. More serious criminal offenses and civil cases with higher stakes are heard by a three judge panel and appealed to a panel of five more senior judges. There are some variations from country to country, but often an extremely serious criminal offense will be heard in the first instance by a mixed panel of three senior judges and significantly more "lay judges" who are a bit like jurors, or by a panel of five senior judges. In some countries extremely weighty civil cases are heard in the first instance by a panel of five senior judges. In civil law systems, there is often a second level of appeal that like a common law appeal only considers the law and does not receive new evidence.
In the U.S., judges are generally in a second career after a successful career as a lawyer (often as a prosecutor but not exclusively), and are usually politically connected and tend to be very politically aware. Higher level judicial posts are frequently filled by people who have not previously held lower level judicial posts.
In the U.K. and in civil law countries, judges generally choose that career early on in their working life, and progress up the ranks from lower courts to higher courts as civil servants.
3. Contempt of Court.
In a common law system, a judge has contempt of court power. One part of this power is the power to summarily, without a trial, punish people who in the court's presence disrupt its proceedings or show disrespect to the court with a fine and/or incarceration. Another part of this power is to fine or incarcerate someone who is not complying with a court order, either by fining and/or incarcerating them until they comply, or by issuing a fixed fine and/or period of incarceration for violating it (e.g. when compliance is no longer possible).
This power is used to give judges broad power to issue injunctive relief, i.e. to order them to do something or refrain from doing something.
In civil law systems, judges don't have broad contempt of court power. Misconduct in court is dealt with through the criminal justice system just like any other setting. Injunctive relief power in civil law systems is more limited.
4. Judicial Review.
In the U.S., all judges have the right and the obligation to rule on the constitutionality of laws and government action in cases tried before them. (This is not the case in the U.K.)
In most civil law systems, constitutional violations can only be raised in a separate constitutional court.
5. Sources Of Law.
In common law countries, most of the core principles of private law are determined through interpretation of case law precedents with no statutory foundation. Case law precedents are also pivotal in interpreting other sources of law like statutes and a constitution.
In civil law countries, the core principles of all areas of law are codified in statutes, and the role of precedents in interpreting them is smaller.
6. Public law.
In common law systems, the judges who handle cases involving public law disputes between private persons and the government, and between governmental entities are predominantly handled by ordinary senior judges who also handle serious civil and criminal cases.
In civil law systems, public law cases are usually handled in a separate legal system (which is one of the factors that make their ordinary courts less political) and ordinary judges have little or no authority over governmental officials in their official capacity outside criminal cases.
For example, in France, if you have a dispute with the government, you fill out a simple form available at your local convenience store and send it to the Council of State. It assigns senior public law lawyers to both you and the state and they litigate the matter in a very different procedure than ordinary courts. The Council of State, if it identifies a systemic problem will insist that the agency reform itself not just in the case identified but in all cases arising from the systemic problem.